Local Class Field Theory

Home , Discrete valuation, Field norm, Local class field theory, Local field

Richard Crew

April 6, 2019 2 Contents

1 Nonarchimedean Fields 5 1.1 Absolute values ...... 5 1.2 Extensions of nonarchimedean ﬁelds ...... 22 1.3 Existence and Uniqueness theorems...... 37

2 Ramification Theory 49 2.1 The Different and Discriminant ...... 49 2.2 The Ramification Filtration ...... 60 2.3 Herbrand’s Theorem ...... 64 2.4 The Norm ...... 70

3 Group Cohomology 77 3.1 Homology and Cohomology ...... 77 3.2 Change of Group ...... 89 3.3 Tate Cohomology ...... 99 3.4 Galois Cohomology ...... 109

4 The Brauer Group 113 4.1 Central Simple Algebras ...... 113 4.2 The Brauer Group ...... 121 4.3 Nonarchimedean Fields ...... 136

5 The Reciprocity Isomorphism 143 5.1 F -isocrystals ...... 143 5.2 The Fundamental Class ...... 154 5.3 The Reciprocity Isomorphism ...... 160 5.4 Weil Groups ...... 167

6 The Existence Theorem 175 6.1 Formal Groups ...... 175 6.2 Lubin-Tate Groups ...... 184 6.3 The Lubin-Tate Reciprocity Law ...... 191

3 4 CONTENTS Chapter 1

Nonarchimedean Fields

Without explicit notice to the contrary, all rings have an identity and all homomorphisms are unitary, i.e. send the identity to the identity.

1.1 Absolute values

1.1.1 Deﬁnitions. Let R be a ring (not necessarily commutative). An absolute value on R is a map R → R≥0, written x 7→ |x|, with the following properties:

|x| = 0 if and only if x = 0. (1.1.1.1) |xy| = |x||y|. (1.1.1.2) |x + y| ≤ |x| + |y|. (1.1.1.3)

We are mostly interested in the case when R is a ﬁeld or a division ring, but the general notion will be useful at times. We will see numerous examples in section 1.1.4 and elsewhere. For now we make the following elementary observations:

• 1.1.1.2 implies |1| = | − 1| = 1, so that | − x| = |x| for all x ∈ R. Then 1.1.1.3 is equivalent to the triangle inequality: |x − y| ≤ |x| + |y|.

• If x ∈ R× then 1.1.1.1 and 1.1.1.2 imply |x−1| = |x|−1.

• If R is a division ring and | | is map satisfying 1.1.1.2, then 1.1.1.1 is equivalent the condition that | | is not identically zero.

• An absolute value on R deﬁnes an absolute value on the opposite ring Rop.

• If R0 ⊆ R is a subring, an absolute value on R induces one on R0.

• The image |R×| of the unit group is a subgroup of R>0, the value group of the absolute value. We will denote it by ΓR.

5 6 CHAPTER 1. NONARCHIMEDEAN FIELDS

Most of the absolute values in this book have the property of being nonar- chimedan, which means that condition 1.1.1.3 is replaced by the following stronger condition: |x + y| ≤ max(|x|, |y|) for all x, y ∈ R. (1.1.1.4) An absolute value that is not nonarchimedean is called archimedean. If | | is nonarchimedean, the triangle inequality says that |x−y| ≤ max |x|, |y|. From this we deduce the following sharper form of 1.1.1.3:

|x|= 6 |y| ⇒ |x + y| = max |x|, |y|. (1.1.1.5)

Suppose for example |x| < |y|; if |x + y| < |y| then |y| = |x + y − x| ≤ max(|x + y|, |x|) < |y|, a contradiction. We could rephrase 1.1.1.5 by saying that every triangle is isosceles. For a ∈ R and r ≥ 0 the sets

B(a, r) = {x ∈ R | |x − a| ≤ r} (1.1.1.6) B(a, r−) = {x ∈ R | |x − a| < r} (1.1.1.7) (1.1.1.8) are the closed and open ball of radius r about a respectively. Another consequence of 1.1.1.5 is that if two balls are not disjoint then one is contained in the other.

Proof. The condition is clearly necessary. If |x| ≤ |y| and y 6= 0 then |xy−1| ≤ 1, so that |xy−1 + 1| ≤ 1 and thus |x + y| ≤ |y| = max(|x|, |y|).

1.1.2 The integer ring and its ideals. If | | is a nonarchimedean valuation on R, the set

OR = {x ∈ R | |x| ≤ 1} is a subring of R; if R is a ﬁeld it is an integral domain. In any case it is called the integer ring of the absolute value. If K is a division ring, the unit group of OK is × OK = {x ∈ K | |x| = 1}. (1.1.2.1) × For example if x ∈ K satisﬁes |x| < 1 then 1 + x ∈ OR . For any ring R the sets

Ir = {x ∈ R | |x| ≤ r} (1.1.2.2) I

1.1.2.1 Proposition Suppose K is a division ring with a nonarchimedean absolute value and S ⊆ OK is a nonempty subset. If

r = sup |x|. (1.1.2.3) x∈S the left (resp. right) ideal of OK generated by S is Ir if there is an x ∈ S such that |x| = r, and I

Proof. It suffices to treat the case of left ideals, and in either case the inclusion I ⊆ Ir is clear. Suppose first there is x ∈ I with |x| = r; then y ∈ Ir implies −1 −1 |y| ≤ |x|, so that |yx | ≤ 1, yx ∈ OK and finally y = (y/x)x ∈ I. If there is no x ∈ I such that |x| = r we must have I ⊆ I

1.1.2.2 Corollary Suppose K is a division ring with a nonarchimedean absolute value. Then all left and right ideals of OK are 2-sided, and the set of ideals of OK are totally ordered under inclusion. In particular OK is a local ring with maximal ideal I<1.

The ring OK is not necessarily noetherian, but it is “almost” a PID:

1.1.2.3 Corollary A ﬁnitely generated ideal in OK is principal.

Proof. In fact if S is finite, the maximum in 1.1.2.1 and then I = (x) for any x such that |x| = r. A commutative ring with this property of proposition 1.1.2.3 is called a Bezout ring. Suppose K is a division ring and m = I<1 is the maximal ideal of OK . The quotient OK /m = k is also a division ring and is called the residue field of OK , or of K (it might be a skew field if K is). The reduction map OK → k will generally be written x 7→ x¯, and we will use the same notation for all derived homomorphisms, such as the reduction maps OK [X] → k[X], OK [[X]] → k[[X]] etc. Suppose now K is a field with a nonarchimdean valuation. We say that K is equicharacteristic if K and its residue field k have the same characteristic (note that the notion of characteristic has an obvious meaning for a division ring). If they have different characteristics then K must have characteristic 0 and k characteristic p > 0. In this case we say that K has mixed characteristic. One says “equicharacteristic p” or “mixed characteristic p” to indicate the characteristic. If K is a division ring and a ∈ K× then |axa−1| = |x| for all x ∈ K. Therefore the inner automorphism induced by a induces automorphisms of the integer ring OK and the residue field k. We denote by adk(a) the automorphism of k induced in this way by a ∈ K. If the residue field k is commutative, adk(u) 8 CHAPTER 1. NONARCHIMEDEAN FIELDS

× is trivial for any u ∈ OK , and it follows that adk(a) depends only on |a|. Clearly adk(ab) = adk(a)adk(b), so we have deﬁned a homomorphism

invK :ΓK → Aut(k) (1.1.2.4) for any division ring K whose residue field k is commutative. We could call this map the “invariant map” but this term is already used in a different (but closely related sense) for division algebras over local fields.

1.1.3 Valuations. A nonarchimedean absolute value on a ring R can be given by the function v : R → R ∪ {∞} deﬁned by ( − log x x 6= 0 v(x) = (1.1.3.1) ∞ x = 0 where the logarithm can have any base greater than one. The function v(x) clearly satisﬁes the conditions

v(x) = ∞ if and only if x = 0 (1.1.3.2) v(xy) = v(x) + v(y) (1.1.3.3) v(x + y) ≥ v(x), v(y) (1.1.3.4) and conversely any function R → R ∪ {∞} satisfying the conditions 1.1.3.2- 1.1.3.4 deﬁnes an valuation by taking |x| = a−v(x) for any a > 1. A function R → R ∪ {∞} satisfying 1.1.3.2-1.1.3.4 is called a valuation. A valuation is trivial if this is the case for the associated absolute value, or in other words if v(x) = 0 for all x 6= 0. The subset v(R×) ⊆ R is the value group of the valuation; it is isomorphic to the value group of the associated absolute value. We will use the same notation ΓR for this group. If I = (x) ⊆ OK is a principal ideal we will use the notation v(I) = v(x) and speak of v(I) as the valuation of I. In fact if (x) = (y) then y = ux for some × u ∈ OK , and thus v(x) = v(y).

1.1.3.1 Remark In the deﬁnition the additive group R can be replaced by any totally ordered group, and functions of this sort are also called valuations. With that deﬁnition what we are calling valuations are called rank one valuations. We will not use this more general notion. We leave to the reader the proof of the following easy

1.1.3.2 Lemma A subgroup of the additive group of R is either inﬁnite cyclic or dense.

If R is a ring with valuation v, we say that the valuation v is discrete if the value group ΓR ⊂ R is infinite cyclic, and nondiscrete otherwise. If v is a × discrete valuation on R an element π ∈ OK such that v(π) generates v(R ) is called a uniformizer of OR (or of R). If R is a division ring, any two uniformizers 1.1. ABSOLUTE VALUES 9 differ by a unit factor on the left or right. If K is a division ring and π is a n uniformizer, it follows from proposition 1.1.2.1 that the ideals of OK are (π ) for all n ≥ 0. A valuation v on K is normalized if v(π) = 1 for one (or equivalently, any) uniformizer π. Any discrete valuation can be normalized by dividing it by v(π) for any uniformizer π. When a valuation v of K is given we denote by vK the unique normalized valuation that is a multiple of v. It is clear that if K is a field with a discrete valuation, OK is a discrete valuation ring and is in particular noetherian. Similarly if K is a division ring, OK is (left and right) noetherian; it is as it were a “noncommutative discrete valuation ring.” In general, if v is a nondiscrete valuation on a ring R then OR is not noetherian. In fact in this case 0 must be an accumulation point of v(K×); then 1 is an accumulation point of the corresponding absolute value, and the set of I

1.1.4 Examples. Suppose K is a division ring with an absolute value. The >0 minimal possibility for the value group ΓR is the trivial group 1 ⊂ R , so that ( 1 x 6= 0 |x| = 0 x = 0 and conversely the above formula defines a nonarchimedean absolute value on K. It is called the trivial absolute value. Any absolute value on a finite field is trivial. The field R has an archimedean absolute given by |x| = max(x, −x). Simi- larly the complex norm is an archimedean absolute value on the field of complex numbers C, and the quaternionic norm is an archimedean absolute value on the field H of Hamilton quaternions. The Gelfand-Tornheim theorem asserts that any (commutative) field with an archimedean absolute value is isomorphic to a subfield of C (and of course any such field has an archimedean absolute value). In particular, any field with an archimedean absolute value has cardinality at most that of the continuum. If R is a discrete valuation ring with maximal ideal p and fraction field K, the function vp : K → N ∪ {∞} defined by

n vp(x) = the largest n ∈ Z such that x ∈ p (1.1.4.1) defines a normalized discrete valuation on K such that OK = R. More generally if R is a Dedekind domain with fraction field K, the formula 1.1.4.1 defines a normalized discrete valuation vp on the localization Rp of R at p for any maximal ideal p ⊂ R, and thus a normalized valuation vp on K itself. The integer ring of vp is Rp. A particularly important special case is when K is a number field, i.e. a finite extension of Q, and R is the integer ring of K, i.e. the integral closure of Z in K. When R = Z, K = Q the valuation associated to p = (p) is denoted r by vp and can be computed as follows: if x = p (a/b) with a and b prime to 10 CHAPTER 1. NONARCHIMEDEAN FIELDS

−r p, then vp(x) = p . The integer ring of vp is the localization Z(p). Thep-adic absolute value of Q is the absolute value | |p associated to vp. A number field K also has archimedean absolute values which arise as follows: the tensor product R ⊗Q K is isomorphic to a direct sum of copies of R and C, whence a set of embeddings of K into C. The standard archimedean absolute value of C then induces archimedean absolute values on K. If k is any field and and K is a finite extension of the field k(T ) of rational functions in one variable, the integral closure R of the polynomial ring k[T ] in K is a Dedekind domain (this is easy if k has characteristic 0, and a more difficult theorem of Zariski in general). The above remarks then apply to K, so that fields of algebraic functions have a large supply of discrete valuations. Finally let k be any field and set K = k((T )), the field of Laurent series with P n coefficients in k. For f = n−∞ anT ∈ K,

v(f) = min{n|an 6= 0}. defines a normalized discrete valuation with integer ring k[[T ]] and residue field k; it is called the T -adic valuation of K. This shows that any field can be residue field of a discretely valued field. Fields with nondiscrete valuations can be constructed by taking infinite algebraic extensions of the above examples, as we shall see later.

1.1.5 The topology definded by an absolute value. The axioms 1.1.1.1 and 1.1.1.3 say that an absolute value on R is a metric, whence a Hausdorff topology on R. A basis of neighborhoods of a ∈ K is given by sets of open balls B(a, r−) for all r > 0. The axioms 1.1.1.2 and 1.1.1.3 imply that addition and multiplication are continuous. When R is a field K, division is also continuous since |x − x0| < |x0| implies |x| = |x0| and therefore

1 1 |x − x0| |x − x0| − = = 2 . x x0 |xx0| |x0| The calculation in fact shows that inversion is uniformly continuous on any set of the form r ≤ |x| for r > 0. Thus K is a topological ﬁeld for this topology, and in particular its additive and multiplicative groups are topological groups. By construction the metric is continuouus for the topology it deﬁnes. It follows that the ideals I

Since any closed ball is a disjoint union of closed bass of smaller radius, K is totally disconnected. Two absolute values | |1, | |2 are equivalent if they deﬁne the same topology on K. For example the absolute values | |∞ and | |p on Q are inequivalent 1.1. ABSOLUTE VALUES 11

n n for all p: in fact |p |∞ → ∞ as n → ∞, while |p |p → 0. One can show in the same way that | |p and | |q are inequivalent for distinct primes p, q. A theorem of Ostrowski shows that any absolute value on Q is equivalent to | |∞ or to | |p for some prime p. We will not prove this here. More generally, if R is the integer ring of a number ﬁeld K then any absolute value on K is equivalent to one of the vp constructed in section §1.1.4, or else one of the r1 r2 archimedean absolute values arising from the decomposition R⊗QK ' R ⊕C .

1.1.5.1 Theorem Let K be a division ring and suppose | |1, | |2 are nonarchimedean absolute values on K. The following are equivalent:

(i). | |1 and | |2 are equivalent.

(ii). |x|1 < 1 (resp. |x|1 = 1, |x|1 > 1) if and only if |x|2 < 1 (resp. |x|2 = 1, |x|2 > 1). α (iii). |x|1 = |x|2 for some α > 0. n Proof. (i) implies (ii): The inequality |x|1 < 1 implies that |x|1 → 0 as n → ∞ n and thus x → 0 for the topology defined by | |1. But |x|2 ≥ 1 would imply n that {x }n>0 does not converge to zero if | |2 defines the same topology as | |1. Therefore |x|2 < 1. Since (i) is symmetric in the two absolute values, we have shown that |x|1 < 1 −1 if and only if |x|2 < 1. If |x|1 > 1 then x 6= 0 and |x |1 < 1, and from the preceding argument we conclude that |x|1 > 1 if and only if |x|2 > 1. By elimination, |x|1 = 1 if and only if |x|2 = 1. (iii) clearly implies (i). (ii) implies (iii): It suffices to show that log |a| log |b| 1 = 1 log |a|2 log |b|2 for a, b ∈ K such that both fractions are defined and nonzero. Equivalently, we must show that log |a| log |a| 1 = 2 log |b|1 log |b|2 for all such a, b. If not we may assume that the left hand side is the smaller one and that log |a| m log |a| 1 < < 2 log |b|1 n log |b|2 for some a, b ∈ K and integers m, n. By (ii) we know log |b|1 and log |b|2 have the same sign, and we may further assume n has the same sign as log |b|1 and log |b|2. Then n log |b|1 and n log |b|2 are positive and these inequalities imply n −m n −m |a b |1 < 1, |a b |2 > 1 which contradicts (ii). We say that two valuations of K are equivalent if the corresponding absolute values are. From the theorem and 1.1.3.1 we see that two valuations v1, v2 are equivalent if and only if v1 = αv2 for some α > 0. 12 CHAPTER 1. NONARCHIMEDEAN FIELDS

1.1.6 Completion. Suppose K is a commutative ring with an absolute value. Since addition, multiplication and inversion are all continuous, the extend to the completion Kˆ of K for the metric deﬁned by the absolute value. The absolute value likewise extends to Kˆ since it too is continuous. The closure OˆK of the integer ring OK ⊂ K is the integer ring of Kˆ . For any r ≤ 1 the natural maps OK /Ir → OˆK /Iˆr and OK /I

1.1.6.1 Proposition If R is a discretely valued commutative, the maps

n n fn : OˆR → OˆR/mˆ 'OR/m induce a topological isomorphism

Oˆ ' lim O /mn. (1.1.6.1) R ←− R n

n where each OR/m has the discrete topology.

Proof. It suffices to show that the fn satisfy the universal property of the inverse n limit. Suppose in fact that S is a ring and gn : S → OR/m are a family of homomorphisms compatible with the projection maps. For any x ∈ S and n > 0 n pick xn ∈ OR such that the image of xn in OR/m is gn(x). Then (xn)n>0 is a Cauchy sequence and we define g(x) = limn xn ∈ OˆR. It is easily checked that g is a ring homomorphism and that gn is the composition of g with the projections. A nonarchimedean ring is a (not necessarily commutative) ring R with a nonarchimedean absolute value for which R is complete. Examples of noncommutative nonarchimedean division rings will be constructed later. For nonarchimedean fields, we have the following important examples: For any prime p the field of p-adic numbers is the completion Qp of Q for the p-adic absolute value | |p on Q. The closure of of the valuation ring Z(p) the integer ring of Qp; it is the ring of p-adic integers. For any field k the Laurent series field K = k((T )) is complete with respect to its T -adic valuation, and is thus a nonarchimedean field.

1.1.6.2 Proposition Any nonarchimedean field of mixed characteristic p > 0 is an extension of Qp. Proof. If K is any such field then p ∈ m or equivalently |p| < 1. Then any × integer a prime to p is a unit in OK , for if ra + sp = 1, ra = 1 − sp ∈ OK × r and then a ∈ OK as well. Now any x ∈ Q can be written x = p (a/b) with integers r, a, b and a, b relatively prime to p. Then |x| = |p|−r and the absolute value induced on Q is equivalent to the p-adic one. The closure of Q in K is isomorphic to Qp as a nonarchimedean field, so K is an extension of Qp. Convergence questions in nonarchimedean rings are particularly simple. 1.1. ABSOLUTE VALUES 13

1.1.6.3 Proposition A sequence {an}n≥0 in a nonarchimedean ring is a Cauchy sequence if and only if limn→∞ |an − an+1| = 0.

Proof. The condition is evidently necessary. Suppose conversely that lim |an − an+1| = 0 that > 0 is given. Pick an integer N such that |an − an+1| < for all n ≥ N; then m > n ≥ N implies

|an − am| = |(an − an+1) + (an+1 − an+2) + ··· + (am−1 − am)|

≤ max(|an − an+1|, |an+1 − an+2|,..., |am−1 − am|) < and thus {an}n≥0 is a Cauchy sequence.

P 1.1.6.4 Corollary A series n≥0 an in a nonarchimedean ring converges if and only if |an| → 0 as n → ∞. The well-known criteria for convergence of power series in the case of K = R or C hold for any power series with coeﬃcients in a commutative nonar- P n pn chimedean ring: if f(X) = n≥0 anX and lim supn→∞ |an| = L exists, set −1 P n R = L if L 6= 0 and R = ∞ otherwise; then the series f(x) = n≥0 anx converges for any x ∈ K such that |x| < R, and diverges if |x| > R. P n For example if f = n anX ∈ OK [[X]] then f(x) is convergent for any x pn n pn such that |x| < 1, since we must have lim supn |anx | ≤ |x| lim supn |an| ≤ |x|. If in fact an → 0 as n → ∞, f(x) converges for |x| ≤ 1 by corollary 1.1.6.4.

1.1.6.5 Binomial series. In any field K of characteristic zero we can define a the binomial coefficient n for a ∈ K by the usual polynomial formula: a a(a − 1) ··· (a − n + 1) = . n n!

a a If a ∈ Zp then n ∈ Zp for all n ≥ 0. In fact since n is a polynomial it is a a continuous function of a and n ∈ Z for any a ∈ Z; then for z ∈ Zp the value a of n must lie in the closure of Z in Qp, which is Zp. We have seen that if K is nonarchimedean and of mixed characteristic p > 0, Qp is canonically a subﬁeld of K, so for any a ∈ Zp, X a (1 + X)a = Xn n n≥0 deﬁnes an element of K[[X]]. The previous remarks show that it coverges in the open disk |X| < 1. The formula

(1 + X)a(1 + X)b = (1 + X)a+b holds for all a, b ∈ Zp since it clearly holds for a, b ∈ Z and on both sides the coeﬃcient of Xn is a continuous function of a and b. Thus if x ∈ K satisﬁes |x| < 1, (1 + x)1/n is an nth root of 1 + x for any integer prime to p. 14 CHAPTER 1. NONARCHIMEDEAN FIELDS

1.1.6.6 Exponentials and logarithms. Still supposing that K is a mixed characteristic nonarchimedean ﬁeld, the series

X Xn log(1 + x) = (−1)n+1 (1.1.6.2) n n≥1 belongs to K[[X]], and the series converges if we substitute for X any x ∈ K satsifying |x| < 1. In fact if the valuation on K is normalized so that v(p) = 1, v(n) ≤ logp n and then since v(x) > 0,

xn v ≥ nv(x) − log n → ∞ n p as n → ∞. The exponential is similarly deﬁned by

X Xn exp(X) = n! n≥0 and the radius of convergence can be found with the root test: since ! r 1 1 n − s(n) 1 v n = − → − n! n p − 1 p − 1 we see that the series for exp(x) converges if v(x) > 1/(p − 1), and diverges if v(x) < 1/(p − 1).

1.1.7 Local Fields. A nonarchimedean ﬁeld K is a local ﬁeld if it is locally compact with its metric topology.

1.1.7.1 Lemma If K is a locally compact division ring, the ideals Ir, I

Proof. If U is a compact neighborhood of 0, Ir ⊆ U for all suﬃciently small r > 0, and any such Ir is compact since it is closed in U. Pick a nonzero r ≤ 1 × such that Ir ⊆ U; then r = |x| for some x ∈ K and multiplication by x induces ∼ a homeomorphism OK −→ Ir and it follows that OK is compact. Since the Ir and I

1.1.7.2 Proposition A nonarchimedean division algebra is locally compact if and only if its valuation is discrete and its residue field is finite. Proof. Let K be such an algebra. To prove the conditions are sufficient it is enough to show that OK is a compact neighborhood of 0. It is certainly a n neighborhood. Since the residue field is finite the quotient rings OK /m are finite for all n. The topological isomorphism 1.1.6.1 then shows that OK is compact. 1.1. ABSOLUTE VALUES 15

Suppose now K is a local field and S is a set of representatives in OK of the residue field k. Then [ OK = (a + m) a∈S is a disjoint union and the a + m are open. It follows that S, and therefore k is finite. If the valuation of K is not discrete, some point r ∈ |K×| must be an accumulation point. Then it is easy to check that there is then a sequence {rn}n≥0 with 0 < rn < r such that limn rn = r. Then [ I

1.1.8 Systems of representatives. Let K be a nonarchimedean ﬁeld with residue ﬁeld k.A system of representatives of k is a splitting s : k → OK of the canonical homomorphism OK → k.

1.1.8.1 Proposition Suppose K is a nonarchimedean ﬁeld with a normalized discrete valuation v, valuation ring OK and residue ﬁeld k = OK /m. Choose a generator π of the maximal ideal m and a section s : k → OK of the reduction map V → k. Then every x ∈ K has a unique expression in the form

X n x = s(an)π (1.1.8.1) n≥N where N = v(x) and each an ∈ k for all k. Proof. If we replace x by xπ−N with N = v(x) we reduce to the case v(x) = 0, and in partcular x ∈ OK . We will choose the ak successively so that for all n ≥ 0, X i n+1 x − akπ ∈ π OK . 0≤k≤n

Recall that the reduction map OK → k is denoted a 7→ a¯, and let a0 = s(¯x), so that x − a0 ∈ πOK . If a0, . . . , an have been determined we set xn = P i n+1 ¯ 0≤k≤k akπ . Then x − xn = π b for some b ∈ OK and we set an+1 = s(b). ¯ n+2 Since b ≡ s(b) mod π, x − xn+1 ∈ π OK . Clearly |x − xn| → 0 as n → ∞, P k yielding 1.1.8.1. To show uniqueness it suﬃces to observe that if x = k≥N akπ P k and y = k≥N bkπ are distinct series of this type we ﬁnd the smallest k such that ak 6= bk. Since ak and bk are in the image of s we must have ak 6≡ bk k mod π. Then x − y ≡ ak − bk mod π , whence x 6= y. 16 CHAPTER 1. NONARCHIMEDEAN FIELDS

1.1.8.2 Remark. In fact if πn ∈ L is any set of elements of L indexed by Z such that v(πn) = n for all k, then any x ∈ L has a unique representation in the form X x = s(an)πn (1.1.8.2) n≥N as can be proven with the same argument as in the proposition. For the field Qp it is conventional to take π = p and s so that its image is the set 0, 1, . . . p − 1. For x ∈ Z the series 1.1.8.1 is the base p expansion of x, so we can think of the proposition as providing a sort of base p expansion for any element of Qp. For the nonarchimedean field K = k((T ) with its T -adic valuation, the ak are just the Laurent series coefficients.

1.1.8.3 Corollary A discretely valued nonarchimedean ﬁeld with residue ﬁeld k has cardinality |k|ℵ0 .

In particular the cardinality of Qp or of Fq((T )) is the that of the continuum. The corollary implies that there are nonarchidean fields of arbitrarily large cardinality. In fact if k is any field the Laurent series field k((T )) is an equicharacteristic nonarchimedean field of cardinality greater than |k|; when k has characteristic p > 0 we will see later how to construct mixed characteristic fields with this property.

1.1.9 Polynomials and their roots. If R is a quotient of OK we will use the notation x 7→ x¯ for the reduction map R → k by a 7→ a¯, and similarly for R[X] → k[X].

1.1.9.1 Lemma Suppose K is a nonarchimedean field with integer ring OK and π is an element of the maximal ideal. If f, g ∈ (OK /πOK )[X] are polynomials whose images in k[X] generate the unit ideal, then f and g generate the unit ideal in (OK /πOK )[X]. ¯ ¯ Proof. Choose a, b ∈ (OK /πOK )[X] such that a¯f + bg¯ = 1. The ideal I ⊆ OK /πOK generated by the coefficients of af + bg − 1 is contained in the maximal ideal and finitely generated; it is therefore nilpotent. If we set M = (OK /πOK )[X]/(f, g) then M/IM = 0, and thus M = 0 by Nakayama’s lemma.

1.1.9.2 Theorem (Hensel’s Lemma) Suppose K is a nonarchimedean ﬁeld with integer ring OK and residue ﬁeld k. Let f ∈ OK [X] be a polynomial such that f¯ = g0h0 with relatively prime polynomials g0, h0 ∈ k[X]. Then there are 0 ¯ 0 0 polynomials g, h ∈ OK [X] such that f = gh, g¯ = g , h = h and deg g = deg g . If g0 is monic we can take g monic as well. Proof. The theorem reduces immediately to the case where g0 is monic. Choose 0 0 g , h ∈ OK [X] so that g¯0 = g0, h¯0 = h0, deg g0 = deg g0, deg h0 ≤ deg f − deg g0 1.1. ABSOLUTE VALUES 17

0 0 0 and g is monic. Since g and h are relatively prime there are a, b ∈ OK [X] such that ag¯ 0 + ¯bh0 = 1. The ideal generated by the coeﬃcients of f − g0h0 and ag0 + bh0 − 1 is ﬁnitely generated and thus principal, say (π). For n ≥ 0 set n+1 On = OK /π OK . We will show by induction that for all n ≥ 0 there are gn, hn in On[X] such that

f 7→ gnhn under OK [X] → On[X]

gn+1 7→ gn under On+1[X] → On[X]

hn+1 7→ hn under On+1[X] → On[X] 0 0 deg gn = deg g and gn is monic if g is 0 deg hn ≤ deg f − deg g . Then g = lim g and h = lim h satisfy the conditions of the theorem. ←−n n ←−n n 0 0 We take g0 and h0 to be the reductions of g and h modulo π. If gn and hn have been found, we look for gn+1, hn+1 of the form

0 n 0 n gn+1 = gn + π u, hn+1 = hn + π v

0 0 where gn, hn are elements of On+1[X] lifting gn and hn, u and v can be viewed as elements of O0[X] and we need deg u < deg g0, deg v ≤ deg f − deg g0 to insure that gn+1 and hn+1 satisfy the last two displayed conditions above. Since n > 1, 2n ≥ n+2 and the condition that f reduce to gn+1hn+1 in On+1[X] is 0 0 n 0 0 n+1 f ≡ gnhn + π (gnv + hnu) mod π since 2n ≥ n + 1. If we set, with an obvious sense, f − g0 h0 w = g0 v + h0 u = n n ∈ O [X] n n πn 0 0 then we have to solve g0v + h0u = w with deg u < deg0 = deg g . Since 0 g and h−1 generate the unit ideal of k[X], g0 and h0, lemma 1.1.9.1 shows that they generate the unit ideal of O0[X] and it follows that we can solve g0v + h0u = w for u and v. Furthermore we are always free to change u and v by u 7→ u−qg0, v 7→ v +qh0 for any q ∈ O0[X]. Since g0 is monic we can choose q so that deg u < deg g0 by Euclidean division. Since deg h0 ≤ deg f − deg g0, deg h0u ≤ deg f and thus deg g0v ≤ deg w ≤ deg f as well and it follows that 0 deg v ≤ deg f − deg g0 = deg f − deg g as required.

1.1.9.3 Corollary With K as in the theorem, suppose f ∈ OK [X] is such that ¯ f has a simple root a ∈ k. Then f has a root in OK reducing to a modulo m. Proof. Apply the theorem to f¯ = (X − a)h0 and observe that X − a and h0 are relatively prime. 18 CHAPTER 1. NONARCHIMEDEAN FIELDS

1.1.9.4 Roots again. Consider for example the polynomial f(X) = Xp−1−1 ∈ Zp[X]. Modulo p, f has p−1 distinct roots 1, 2, . . . , p−1. The corollary implies that Zp has a full set of (p − 1)-st roots of unity, whose reductions modulo p are the p − 1 elements of Fp. When p > 2 Z has no (p − 1)-st roots of unity, which strongly suggests that completeness is essential here. There are nonetheless incomplete discrete valuation rings, in fact fairly important ones, for which the Hensel’s lemma is true. We saw in section 1.1.6.5 that if K is a nonarchimedean ﬁeld of mixed characteristic p > and p 6 |n, any x ∈ K such that |x − 1| < 1 has an nth-root. × Corollary 1.1.9.3 shows more generally that if a ∈ OK reduces to an n-power × × n in k and p 6 |n then a has an nth root in OK . In fact if f(X) = X − a and n f(a0) ≡ 0 mod m, a0 is a simple root of f modulo m and thus X − a has a × root in OK (and thus necessarily in OK .

1.1.9.5 Corollary If f ∈ K[X] is monic and irreducible then f ∈ OK [X] if and only if its constant term is in OK .

Proof. We need only show that the condition is sufficient. If we write f = n n−1 X + a1X + ··· + an it suffices to show that |ai| ≤ |an| for all i. Set a0 = 1 and choose i as large as possible so that |ai| is maximal. If i = 0 or i = n we are done; otherwise 0 < i < n and dividing by ai yields a irreducible g ∈ OK [X] for i i which the coefficient of X is 1, and aj ∈ m for j > i. We may factor g¯ = X h, where h has nonzero constant term. Then Xi and h are relatively prime, and by Hensel’s lemma g factors into polynomials of degree i and n − i, contradicting the irreducibility of g in OK [X]. When the reduction of f has multiple roots Hensel’s lemma is not applicable, but the following version of Newton’s method can sometimes be used. We will need the following version of Taylor’s theorem with remainder. Recall that for any commutative ring with identity R, differentiation can be defined purely formally on the polynomial ring R[X] and and satisfies the usual identities (additivity, Leibnitz rule). Then for any f(X) ∈ R[X] there is a polynomial g(X,Y ) ∈ R[X,Y ] such that

f(X + Y ) = f(X) + f 0(X)Y + Y 2g(X,Y ). (1.1.9.1)

It suﬃces to check this for mononials, in which case it follows from the binomial identity.

1.1.9.6 Theorem (Newton’s method) Suppose K is a nonarchimedean ﬁeld with integer ring OK , f ∈ OK [X], and a0 ∈ OK is such that

f(a0) c = 0 2 < 1. f (a0)

Then f has a root a ∈ OK such that |a − a0| ≤ c. 1.1. ABSOLUTE VALUES 19

Proof. Deﬁne a sequence of an ∈ K by

f(an) an+1 = an − 0 . f (an)

We will show that an converges to a root. In fact we will show by induction that 0 2 2n |an| ≤ 1 and |f(an)/f (an) | ≤ c (1.1.9.2) for all n ≥ 0. Then 2n |an+1 − an| ≤ c (1.1.9.3) which implies that the method converges rather rapidly, say limn an = a. Since |an| ≤ 1 for all n we have |a| ≤ 1, and then 1.1.9.3 implies |an − a0| ≤ c for all n 2n and thus |a − a0| ≤ c. Finally 1.1.9.2 implies |f(an)| ≤ c and thus f(a) = 0. The ﬁrst inequality in 1.1.9.2 and 1.1.9.3 imply |an+1| ≤ 1. The second 0 implies that h = −f(an)/f (an) satisﬁes

2n 0 |h| < c |f (an)|.

Taylor’s formula 1.1.9.1 yields

0 2 2 f(an+1) = f(an) + f (an)h + h g(an, h) = h g(an, h) and thus 2 |f(an+1)| ≤ |h | 0 since g(X,Y ) ∈ OK [X,Y ] and |an| ≤ 1. Again by Taylor’s formula, f (an+1) − 0 0 f (an) is a multiple of h, which has absolute value less than |f (an)|, so that

0 0 |f (an+1)| = |f (an)|.

Putting it all together, we ﬁnd

2 2n+1 0 2 f(an+1) |h| c |f (an)| 2n+1 0 2 < 0 2 < 0 2 = c f (an+1) |f (an)| |f (an) | as required.

1.1.9.7 Roots yet again. We saw in section 1.1.9.4 that for any nonarchimedean ﬁeld K of mixed characteristic p > 0 and integer n prime to p, an element × a ∈ OK is an nth power if and only if it is an nth power residue, i.e. an nth- power modulo m. We now inquire about pnth roots. We will see later that the n question reduces to the existence of p th roots of a ∈ OK such that |a − 1| < 1. pn For such a, a0 = 1 is a root modulo m of the polynomial f(X) = X − a, and Newton’s method shows that f(X) has a root in K if |f(1)/f 0(1)| < 1. This 2n n 2n says that |(1 − a)/p | < 1, so a has a p th root if |a − 1| < |p| . For K = Qp this says that a is a pnth power if a ≡ 1 mod p2n+1. 20 CHAPTER 1. NONARCHIMEDEAN FIELDS

1.1.10 Zeroes of power series. If K is a nonarchimedean ﬁeld and f ∈ OK [[X]], the root test for the convergence of power series shows that the series f(x) converges for any x ∈ m. In this section we show that the problem of ﬁnding solutions of f(x) = 0 reduces to solving a polynomial equation. We start with a version of the division algorithm.

1.1.10.1 Proposition Suppose R is a complete local ring with maximal ideal P n ¯ m. Let f = n≥0 anX ∈ R[[X]] be such that f 6= 0 and let n be the small- × est natural number such that an ∈ R . For any g ∈ R[[X]] there are unique elements q ∈ R[[X]] and r ∈ R[X] such that

g = qf + r, and deg r < n.

Proof. (Manin) We ﬁrst prove uniqueness. Denote by h : R[[X]] → R[X], t : R[[X]] → R[[X]] the R-linear maps

X k X k X k g = bkX ⇒ h(g) = bkX , t(g) = an+kX . k≥0 0≤k

Then t(Xng) = g for any g ∈ R[[X]], and h(g) = g if g is a polynomial of degree less than n. For f as in the proposition, t(f) ∈ R[[X]]× and the coeﬃcients of h(f) generate a principal ideal (π). Therefore the map

h(f) m (h) = h f t(f) is well-deﬁned and takes its values in πR[[X]]. There are q, r satisfying the conditions of the proposition if and only if t(g) = t(qf). Since f = h(f) + Xnt(f), ﬁnding q and r is equivalent to solving

t(g) = t(q(h(f)) + Xnt(f)) = t(q(h(f))) + qt(f) for q. If we put u = qt(f) this is is the same as

h(f) t(g) = t u + u = (I + t ◦ m )(u). t(f) f

Since π is topologically nilpotent, so is the linear operator mf and the operator −1 −1 I+mf is invertible. We ﬁnd that u = (I+mf ) (g) and q = t(f) u; this proves −1 −1 uniqueness, and conversely if we set q = t(f) (I + mf ) (g) and r = g − qf then q and r satisify the conditions of the proposition.

1.1.10.2 Theorem (Weierstrass factorization) Suppose R is a complete local P n ¯ ring with maximal ideal m. Let f = n≥0 anX ∈ R[[X]] be such that f 6= 0 × and let n be the smallest natural number such that an ∈ R . Then f = gh where g is a monic polynomial of degree n whose coeﬃcients of degree less than n lie in m, and g ∈ R[[X]]×. 1.1. ABSOLUTE VALUES 21

Proof. Apply proposition 1.1.10.1 with f and g = Xn, so that Xn = qf +r with P n deg r < n. If q = n≥0 cnX then comparing the coeﬃcients of both sides of Xn = qf + r shows that the coeﬃcients of r all lie in m, and that X 1 = aicj ≡ anc0 mod m. i+j=n

Then q ∈ R[[X]]× and f = (Xn − r)q−1, and we may take g = Xn − r and h = q−1.

1.1.10.3 Corollary Suppose K is a discretely valued nonarchimedean ﬁeld and f ∈ OK [[X]]. Then f has at most ﬁnitely many zeroes in m.

Proof. By dividing f by a power of a uniformizer we may assume that f¯ 6= 0. If f = gh is the factorization of f given by the theorem, any zero of f is a zero of g and conversely.

1.1.11 Exercises.

1.1.11.1 Let R be a ring with a nonarchimedean absolute value. Show that if two closed balls in R intersect then one is contained in the other. Speciﬁcally, show that if r1 ≤ r2 and B(a1, r1) ∩ B(a2, r2) 6= ∅ then B(a1, r1) ⊆ B(a2, r2).

1.1.11.2 Show that for any prime p and n ∈ N

X n n − s(n) v (n!) = = p pk p − 1 k≥1 where vp is the p-adic valuation of Q and s(n) is the sum of the base p digits of n. In other words, if

2 r n = a0 + a1p + a2p + ··· + arp then

s(n) = a0 + a1 + a2 + ··· + ar. Conclude from this that

n + m s(n) + s(m) − s(m + n) v = . p m p − 1

a 1.1.11.3 Find a formula for vp n when a ∈ Qp \ Zp.

1.1.11.4 Prove lemma 1.1.3.2. 22 CHAPTER 1. NONARCHIMEDEAN FIELDS

1.1.11.5 Show that the polynomial

f(X) = X3 + X2 − 2X + 8 is irreducible in Q[X] but has three roots in Q2. Find the 2-adic valuations of these roots and compute their 2-adic expansions through the term in 26. (Compute f(1), f(2) and f(4)).

1.1.11.6 The Cartesian equation for the lemniscate r2 + cos 2θ = 0 is

(x2 + y2)2 − y2 + x2 = 0.

Let k be any ﬁeld of characteristic not equal to 2. Set K = k((x)) and write the above equation as F (y) = 0 with F ∈ K[y]. Show that F has four solutions in K, with leading terms ±1, ±x, and compute these through the term in x3. When k = R, interpret this in terms of the graph of the lemniscate.

1.2 Extensions of nonarchimedean ﬁelds

1.2.1 Extensions. Let K be a valued field. A valued extension of K is a K-algebra R with an absolute value that extends that of K. If K is nonarchimedean, a nonarchimedean extension of K is valued extension that is complete for its absolute value. In this chapter we are mainly interested in field extensions, but the case where R is a division ring will be important later. If K is a valued field with residue field k and R is a valued K-algebra, the structure homomorphism R → K induces a homomorphism OK ⊆ OL. As this homomorphism sends mK → mR, it induces a homomorphism k → kR, where kR is the residue field of R. In the case where R is a division ring we call kR/k the residual extension of R/K. An extension of discrete valuation rings is an injective homomorphism A → B of discrete valuation rings such that if mA ⊂ A, mB ⊂ B are the maximal ideals then A ∩ mB = mA. Thus if L/K is an extension of discretely valued fields then OK → OL is an extension of discrete valuation rings. Conversely of A → B is an extension of discrete valuation rings and K (resp. L) is the fraction field of A (resp. B), the valuations of A and B induce valuations on K and L. Multiplying the valuation of L by a suitable factor makes it an extension of the valuation of K, and then L/K is an extension of valued fields in the above sense. If K is a valued field and L/K is an field extension, the valuation of K extends to L under fairly general circumstances. We will be particularly concerned with the case when K is nonarchimedean (i.e. complete) and L/K is finite.

1.2.2 Normed spaces. Let K be a ﬁeld with a nonarchimedean absolute value | | and let V be a K-vector space. A norm for V is a map V → R, 1.2. EXTENSIONS OF NONARCHIMEDEAN FIELDS 23 written v 7→ ||v|| such that

||v|| = 0 if and only if v = 0 (1.2.2.1) ||av|| = |a| ||v|| for all a ∈ K, v ∈ V. (1.2.2.2) ||v + w|| ≤ max(||v||, ||w||) for all v, w ∈ V. (1.2.2.3)

As is the case for the absolute value of K, a norm on a K-vector space defines a metric for which the operations of addition and scalar multiplication are continuous. In particular for any v ∈ V , the function x 7→ x + v (translation by v) is a homeomorphism. A Banach space over K is a K-vector space V with a norm such that V is complete for the metric defined by the norm. A K-vector space of finite dimension always has a norm. For example if V has dimension n and v1, . . . , vn is a basis for V it is easily checked that

||v|| = max(|a1|,..., |an|) if v = a1v1 + ··· + anvn (1.2.2.4) defines a norm on V . We will call 1.2.2.4 the standard norm associated to the basis v1, . . . , vn. If K is a nonarchimedean field then V is a Banach space for this norm. Two norms on a K-vector space V are equivalent if they define the same topology. This will be the case in particular if the norms || ||, || ||1 satisfy the inequalities

C||v||1 ≤ ||v|| ≤ D||v||1 (1.2.2.5) for some nonzero constants and all v ∈ V .

1.2.2.1 Proposition Suppose K is a nonarchimedean field and V is a K-vector space of finite dimension. Any norm on V is equivalent to the norm defined by a basis, and V is complete for this norm.

Proof. The assertion is vacuous if V has dimension zero. If V has dimension one with basis v1 then V 1.2.2.5 holds with C = D = ||v1||, and V is evidently complete. Suppose V has dimension n > 1 with basis v1, . . . , vn and the lemma has been proven for K-vector spaces of dimension n − 1. Let || || be a norm on V and denote by || ||1 the norm associated to the basis v1, . . . , vn. For v ∈ V we P write v = i aivi; then X ||v|| = || aivi|| ≤ max(|ai| ||vi||) ≤ max(|ai|) max(||vi||) = ||v||1 max(||vi||) i i i i i and we can take D = maxi(||vi||) in 1.2.2.5. For the other inequality we denote by Vi for 1 ≤ i ≤ n the span of the vj for all j 6= i; then vi 6∈ Vi implies 0 6∈ vi + Vi. For the rest of the argument, topological assertions are relative to the topology deﬁned by || ||. By induction Vi is complete for the induced 24 CHAPTER 1. NONARCHIMEDEAN FIELDS

metric and therefore closed in V . Since translation is a homeomorphism vi + Vi is closed for all i, and [ (vi + Vi) 1≤i≤n is a closed subset of V not containing 0. It is therefore disjoint from some open ball, say ||x|| < C. Let v ∈ V be any nonzero element and write v = P −1 i aivi. Choose i such that ||v||1 = |ai|; then ai v ∈ vi + Vi and consequently −1 |ai| ||v|| ≥ C, i.e. ||v|| ≥ C|ai| = C||v||1 as required.

1.2.2.2 Corollary Suppose K is a nonarchimedean field and V is a K-vector space of finite dimension. Then any two norms on V are equivalent, and for the metric defined by any norm, V is complete.

1.2.2.3 Lemma Let K be a ﬁeld and D a division K-algebra. Any x ∈ D is contained in a subﬁeld of D containing K.

Proof. Consider the K-algebra homomorphism f : K[X] → D sending g(X) to g(x). If x is algebraic over K then the minimal polynomial g of x is irreducible (same proof as for fields) and the image of f is isomorphic to the field K[X]/(g). Otherwise x is transcendental and f is injective. Then f extends to a K-algebra homomorphism K(X) → D, whose image is a field containing K and x. The lemma shows that for any element x of a division K-algebra D it makes sense to speak of the subfield of D generated by x.

1.2.2.4 Theorem Suppose K is a nonarchimedean field and D is a division K-algebra of finite degree over K. the absolute value of K extends uniquely to D, and D is complete for topology defined by the extension. Thus D is naturally a nonarchimedean division ring.

Proof. We first show that the extension is unique if it exists. Suppose | |1, | |2 are two extensions of the absolute value | | of K. Since they are norms on D when viewed as a K-vector space the last corollary says they induce the α same topology and D is complete. Then | |2 = | |2 for some α > 0, and since | |1, | |2 coincide on K ⊆ D we must have α = 1. × For the proof of existence we set n = [L : K] and denote by ND/K : D → × K the norm for the finite-dimensional K-algebra D (i.e. ND/K (x) is the determinant of the K-linear endomorphism of D induced by left multiplication by x. will show that 1/n |x|D = |ND/K (x)| (1.2.2.6) defines an absolute value on D. Corollary 1.2.2.2 shows that D will be complete for the metric defined by | |D. 1.2. EXTENSIONS OF NONARCHIMEDEAN FIELDS 25

It is clear that |x| = 0 if and only if x = 0, and the multiplicative property of the norm shows that |xy|D = |x|D|y|D. To prove 1.1.1.4 it suffices by lemma 1.1.1.1 to show that |x + 1|D ≤ 1 whenever |x|D ≤ 1. We first consider the case where D = L is a field. Pick x ∈ L and let n n−1 f(X) = X + a1X + ··· + an ∈ K[X] be the minimal polynomial of x over K. Up to a sign the characteristic polynomial of multiplication by x is a power of f(X), and by definition NL/K (x) is the constant term of the characteristic d polynomial. We therefore have NL/K (x) = ±(an) for some d (in fact the degree of L/K(x). Thus if |x|D ≤ 1, |an| ≤ 1 as well and by corollary 1.1.9.5, f(X) ∈ OK [X]. Since f(X) is monic irreducible in K[X], g(X) = f(X − 1) is monic irreducible as well and belongs to OK [X] since f(X) does. In fact g(X) is the minimal polynomial of x + 1 over K, and reversing the above argument shows that NL/K (x + 1) ∈ OK , whence |x + 1|D ≤ 1. The theorem is now proven in the case of a finite extension of fields. In the general case D lemma 1.2.2.3 says that D is a union of fields of finite degree over K. Suppose x ∈ D and F = K(x) be the subfield of D generated by x. Then [D:F ] ND/K (x) = NF/K (ND/F (x)) = (NF/K ) and therefore 1/[F :K] |x|D = |NF/K | = |x|F where |x|F denotes the unique extension of the absolute value of K to F . Since we have shown that | |F is an absolute value on F , |x + 1|D ≤ |x|D and | |D is an absolute value on D.

For later use we record the formula for the corresponding valuation vD: v(x) v (x) = . (1.2.2.7) D [D : K]

1.2.2.5 Corollary If L/K is an algebraic extension of a nonarchimedean field K, the absolute value of K extends uniquely to L/K. Proof. The theorem says that the absolute value extends uniquely to every finite extension M/K where M ⊆ L. For any x ∈ L, the absolute value extends to K(x) and serves to define |x|. To check the axioms it’s enough to note that for all x, y ∈ L, M = K(x, y) is a finite extension of K, so the absolute value extends to M. It corollary 1.2.2.5 is not asserted that L is complete or that L is a nonarchimedean field; in fact this is frequently false. For example the algebraic closure ¯ Qp of Qp has a unique absolute value extending the p-adic absolute value of Qp; it is not complete for this absolute value (exercise 1.2.9.2). ¯ The completion of Qp is denoted by Cp. We will see later that Cp is alge- brabraically closed (corollary 1.2.8.50. ¯ The valuation of Qp or Cp is not discrete. In fact for any n > 1 the polynomial f(X) = Xn − p is irreducible in Q[X]. Denote by v the extension of the 26 CHAPTER 1. NONARCHIMEDEAN FIELDS

¯ ¯ p-adic valuation of Q to Qp and let α be a root of f in Qp. Then v(α) = 1/n, and we conclude that the value group of v contains Q ⊂ R. Then formula 1.2.2.7 shows that the value group is exactly Q. Note that a similar argument can be made for any discretely valued nonarchimedean ﬁeld. Because of corollary 1.2.2.5 we may for any extension L/K speak of the absolute value of any x ∈ L that is algebraic over K.

1.2.2.6 Corollary If Kalg /K is an algebraic closure of K and x ∈ Kalg then all the conjugates of x have the same valuation. Proof. If x0 ∈ Kalg is conjugate to x, there is an automorphism σ : Kalg → Kalg such that σ(x) = x0. Since x 7→ |x| and x 7→ |σ(x)| are both absolute values of Kalg extending that of K, we must have |σ(x)| = |x| and therefore |x0| = |x|.

1.2.2.7 Proposition If L/K is an algebraic extension of nonarchimedean fields, OL is the integral closure of OK in L. Proof. Denote the integral closure by A. If x ∈ A satisfies |x| > 1, |x|n+1 > |x|n for all n ≥ 0. On the other hand if x is a root of a monic equation of degree d d i with coefficients in OK then |x| ≤ max0≤i

1.2.3 Residual degree and ramification index. Suppose L/K is an extension of valued fields. The degree of the residual extension is called the residual degree of L/K and is denoted by fL/K . Furthermore ΓK is a subgroup of ΓL and the index [ΓL :ΓK ] is called the ramification index of L/K and is denoted by eL/K If L/K and M/L are extensions of nonarchimedean fields the formulas

eM/K = eM/LeL/K , fM/K = fM/LfL/K . (1.2.3.1) follow immediately from the deﬁnitions.

1.2.3.1 Proposition If L/K is a finite extension of valued fields, eL/K and fL/K are finite and eL/K fL/K ≤ [L : K]. (1.2.3.2)

Proof. Pick integers r ≤ eL/K and s ≤ fL/K ; it is enough to show rs ≤ [L : K]. × By hypothesis there are x1, . . . , xr ∈ L such that |xi| have distinct images in ΓL/ΓK , and y1, . . . , ys ∈ OL with linearly independent images in kL. Note that × this implies that yi ∈ OK for 1 ≤ i ≤ s. We will show that the xiyj are linearly independent over K. Suppose to the contrary that X ai,jxiyj = 0 (1.2.3.3) i,j 1.2. EXTENSIONS OF NONARCHIMEDEAN FIELDS 27

with ai,j ∈ K. Choose (k, `) such that

1.2.3.2 Corollary If L/K is a ﬁnite extension of valued ﬁelds, the valuation of L is discrete (resp. trivial) if and only if the valuation of K is discrete (resp. trivial).

Proof. Since ΓL/ΓK is finite the assertion is clear in the discrete case. In the case of trivial valuations it suffices to observe that the value group cannot be a nontrivial finite group.

1.2.4 Unramified extensions. For the rest of this chapter we will be concerned with finite extensions of valued fields. A finite extension L/K of valued fields is residually separable if the residual extension is separable. A finite extension L/K of valued fields is unramified if it is residually separable and fL/K = [L : K]. The definition implies eL/K = 1, but in general the condition fL/K = [L : K] is stronger.

1.2.4.1 Lemma Let K be a valued field and f ∈ OK [X] a monic polynomial whose reduction f¯ is irreducible and separable. Then L = K[X]/(f) is an unramified extension of K and if x ∈ L is a root of f in L, OL = OK [x] and L = K(x). Proof. Observe first that f is irreducible in K[X] by Gauss’s lemma, so L/K is a field extension of degree deg f. We can assume that x is the image if X ∈ K[X] in L, and it suffices to show that OL = OK [x]. In fact if this is true, the residual ¯ field kL of L is k[X]/(f) where as usual k is the residue field of K; then kL/k ¯ ¯ is separable since f is, and [L : K] = deg f = deg f = [kL : k]. Since K is complete, OL is the integral closure of OK in L; therefore x ∈ OL and OK [x] ⊆ OL. Conversely if y ∈ OL we can write n−1 y = an−1x + ··· + a1x + a0 28 CHAPTER 1. NONARCHIMEDEAN FIELDS

with ai ∈ K and n = deg(f) = [L : K]. We must show that |ai| ≤ 1. If not the suppose the maximum of |a0|,..., |an−1| is attained by |ai| > 1; then dividing the equation by ai and reducing modulo m yields a nontrivial dependence relation among the x¯i with 0 ≤ i < n. This is impossible since x¯ has degree n over k, and it follows that OL = OK [x].

1.2.4.2 Theorem Suppose K is a nonarchimedean field with residue field k and k0/k is a finite separable extension of k.

(i). There is an unramiﬁed extension L/K, unique up to K-isomorphism such whose residual extension is k0/k.

0 0 (ii). If x ∈ L is such that x¯ ∈ k generates k /k then L = K(x) and OL = ¯ OK [x]. If f ∈ OK [X] is the minimal polynomial of x then f is the minimal polynomial of x¯. Conversely if f is any monic lifting of the minimal ¯ 0 polynomial of f then f has a root x ∈ OL such that x¯ generates k /k.

(iii). Suppose E/K is an algebraic extension with residue ﬁeld kE. Any k- 0 homomorphism k → kE extends uniquely to a K-homomorphism L → E.

Proof. Suppose x¯ ∈ k0 generates k0 as an extension of k, f¯ is the minimal poly- 0 ¯ nomial of x over k and choose a monic lifting f ∈ OK [X] of f. By construction f is separable, and it is irreducible by Gauss. The existence of L/K then follows from lemma 1.2.4.1, as does assertion (ii). The uniqueness assertion in (i) follows from (iii), applied to a pair of unramified finite extensions L/K, L0/K with isomorphic residual extensions k0/k and the identity of k0. Suppose then that E/K is an algebraic extension with with residual extension kE/k, and recall that the absolute value of K extends uniquely to E. If x¯, f¯ and f are as before, we can identify L with K[X]/(f). 0 0 Let y¯ be the image of x¯ under the k-homomorphism k → kE and let y ∈ OE be an element lifting y¯. Since E/K is algebraic, K(y) is a finite extension of K and is therefore complete for the absolute value of K(y0) induced by that of E. 0 The residue field of K(y ) is a subfield of kE containing y¯, so by the corollary to Hensel’s lemma the separable polynomial f has a root y ∈ K(y0) The map K[X] → L sending X 7→ y induces an injective homomorphism K[X]/(f) → L since f is irreducible. This is the desired homomorphism L → E; it is evidently unique, since it is determined by the image of x ∈ L, which is in turn determined 0 by the k-homomorphism k → kE.

1.2.4.3 Corollary Two unramiﬁed extensions of K are isomorphic if and only if their residual extensions are isomorphic.

1.2.4.4 Corollary Suppose L/K is a finite unramified extension of nonarchimdean fields with residual extension kL/k. Then any k-automorphism of kL is the reduction of a unique K-automorphism of L. 1.2. EXTENSIONS OF NONARCHIMEDEAN FIELDS 29

1.2.4.5 Corollary A local field has up to isomorphism exactly one unramified extension of any given degree. Proof. Since the residue field of a local field is finite, this follows from the corresponding assertion for finite fields.

1.2.4.6 Proposition Suppose L/K is an finite unramified extension of nonarchimedean fields and F/K is an extension of nonarchimedean fields. Then L ⊗K F is a direct sum of unramified extensions of K.

Proof. By theorem 1.2.4.2 we can write L = K(x) with x ∈ OL integral over ¯ K, and if f ∈ OK [X] is the minimal polynomial of x then f ∈ kX is irreducible ¯ Q and separable. Let f/k be the residual extension of F/K and let f = i g¯i the factorization of f¯ in f[X] into irreducible monic polynomials. Since f¯ ∈ k[X] is separable, the g¯i are separable and relatively prime to each other. By Hensel’s Q lemma we have f = gi with gi ∈ OF [X], gi irreducible and separable and i L reducing modulo mF to g¯i. If we set Li = F [X]/(gi) then L ⊗K F ' i Li, and lemma 1.2.4.1 shows that each Li is an unramiﬁed extension of F .

1.2.4.7 Corollary Suppose K is a nonarchimedean field, E/K is an extension and L, F are finite extensions of K contained in E. (i). If E/L and L/K are finite then E/K is unramified if and only if E/L and L/K are unramified. (ii). If L/K is unramified then so is LF/F . (iii). If L/K and F/K are unramified, so is LF/K. Proof. The first assertion follows from the transitivity of separability and the multiplicative properties of the degree and the residual degree 1.2.3.1. The third assertion follows from the second. As to the second, LF is the image of the natural map L ⊗K F → E, which must be one of the direct summands of L ⊗K F . We will say that an algebraic extension L/K of nonarchimedean fields is unramified if every finite extension of K contained in L is unramified in the previous sense. The first part of corollary 1.2.4.7 shows that this definition is consistent with the previous one. Corollary 1.2.4.7 implies that if L/K is any algebraic extension, the compositum of all finite unramified extensions of K contained in L is unramified; it is called the maximal unramified extension of K in L. If L is an algebraic closure of K it is called “the” maximal unramified extension of K since any two such extensions are isomorphic.

1.2.4.8 Corollary Suppose K is a nonarchimedean field, L/K is an unramified finite extension and F is the maximal unramified extension of K in L. If fL/K is finite then fL/K = [F : K] 30 CHAPTER 1. NONARCHIMEDEAN FIELDS

Proof. It suﬃces to observe that F and L have the same residue ﬁeld.

1.2.5 Totally ramified extensions. An extension L/K of valued fields is tamely ramified if eL/K is finite and not divisible by the residual characteristic, and totally ramified if fL/K = 1.

1.2.5.1 Proposition Suppose K is a discretely valued nonarchimedean field and L/K is a finite totally ramified extension. If τ ∈ L is a uniformizer, OL = OK [τ] and τ is a root of an Eisenstein polynomial in OK [X] of degree eL/K = [L : K]. In particular OL is a free OK -module of rank [L : K]. Proof. Observe first that a system of representatives of K is also one for L. If π i j be a uniformizer of K and v is the normalized valuation of L, v(τ π ) = jeL/K +i and it follows from proposition 1.1.8.1 and the remark following it that any x ∈ OL has a unique expansion in the form

X X i j x = s(ai,j)π τ i≥0 0≤j

e−1 with ai,j ∈ k and e = eL/K . It follows that 1, τ, . . . , τ is a basis of OL as OK -module. In particular OL = OK [τ], L = K(τ) and [L : K] = eL/K . The minimal polynomial f of τ over K is a monic polynomial in OK [X]; since all conjugates of τ have absolute less than one, the coefficients of f lie in m. The constant term is up to a sign the product of the conjugates; there are e of them and they all have the same absolute value, namely |τ| = |π|1/e. The constant term thus has absolute value |π|, so it is a uniformizer in K, and f is an Eisenstein polynomial. Corollary 1.2.4.7 has no analogue for totally ramified extensions: there are totally ramified extensions L/K, F/K with L, F ⊂ E such that LF/K is not totally ramified (c.f. exercise 1.2.9.6). Let M/K be any extension of fields and suppose L, F are subfields of M containing K. Recall that L and F are linearly disjoint if the canonical map L ⊗K F → M is injective; if so its image is the compositum LF of L and F .

1.2.5.2 Proposition Suppose K is a discretely valued nonarchimedean field and L, F are extensions of K contained is some algebraic closure of K. If L/K is unramified an F/K is totally ramified, L and F are linearly disjoint over K. Proof. Since tensor products commute with inductive limits we may assume that L and F are finite extensions of K. It suffices to show that L ⊗K F is a field. By proposition 1.2.5.1 there is an isomorphism F ' K[X]/(f) for some Eisenstein polynomial f ∈ K[X]. Then

L ⊗K F ' L[X]/(f) and since L/K is unramiﬁed, f is an Eisenstein polynomial in L[X]. Therefore L ⊗K F is a ﬁeld. 1.2. EXTENSIONS OF NONARCHIMEDEAN FIELDS 31

n 1.2.5.3 Example: cyclotomic ﬁelds. Denote by µpn the set of p th roots of ¯ 1 in Qp and let Qp(µpn ) be the ﬁeld obtained by adjoining the elements of n µpn to Qp. If ζ is a primitive p th root of 1, i.e. a generator of µpn then Qp(µpn ) = Qp(ζ), and ζ is a root of the polynomial

n Xp − 1 f(X) = Xpn−1 − 1 (1.2.5.1) n−1 n−1 n−1 = X(p−1)p + X(p−2)p + ··· + Xp + 1

Since f(X+1) is an Eisenstein polynomial in Zp[X] f(X) is irreducible, Qp(µpn ) n−1 is totally ramiﬁed of degree (p − 1)p over Qp, and ζ − 1 is a uniformizer of pn−1 Qp(ζ). Furthermore η = ζ is a primitive pth root of 1, so that Qp(η) is an extension of degree p − 1 contained in Qp(µpn ). Thus Qp(η) is the maximal tamely ramiﬁed extension of Qp in Qp(µpn ).

1.2.6 Separable extensions. When L/K is a separable extension of nonarchimedean fields, equality in 1.2.3.3 follows from standard commutative algebra, as we now explain. Let A be any Dedekind domain with fraction field K, L/K a finite separable extension of K and B the integral closure of A in L. Since L/K is separable, B is a finite A-algebra. Let us recall the proof: the relative trace TrL/K : L → K induces a K-valued bilinear form hx, yi = TrL/K (xy) which is nondegenerate when L/K is separable. If we choose a basis of L/K contained in B, B is contained in the span of the dual basis, and is therefore finitely generated. Suppose now m is a maximal ideal of A, and recall that the localization Am is a discrete valuation ring, K is the fraction field of Am and the valuation of Am induces a discrete nonarchimedian valuation of K with integer ring Am. The localization Bm has finitely many maximal ideals p1,... pr lying above m; as before to each pi is associated a discrete valuation of L whose integer ring is the localization Bpi . Denote by ei and fi the ramification index and residual degre of Bpi /Am; the Dedekind degree formula asserts that X [L : K] = eifi. (1.2.6.1) i

We see from this how strict inequality can obtain in 1.2.6.2: since ei, fi ≥ 1 we must have eifi < [L : K] if there is more than one prime ideal of B above m.

1.2.6.1 Theorem If L/K is a separable extension of discretely valued nonarchimedean ﬁelds, eL/K fL/K = [L : K]. (1.2.6.2)

Proof. The preceding discussion applies to A = OK and B = OL since the latter is the integral closure of the former in L. Since K is complete the valuation of K has a unique extension to L. This implies that OL has only one maximal ideal above m, and the equality follows from 1.2.6.1. 32 CHAPTER 1. NONARCHIMEDEAN FIELDS

In section 1.3.2 we will determine all local ﬁelds of positive characteristic. For characteristic zero we have:

1.2.6.2 Theorem A local field of characteristic zero is a finite extension of Qp. Proof. A local field K is discretely valued and the residue field k has characteristic p > 0 for some p. Since K has characteristic 0 it is an extension of Qp by proposition 1.1.6.2. Since K is discretely valued it has finite ramification index over Qp, and since k is finite the residue degree is finite. Since K/Qp is separable, 1.2.6.2 shows that [K : Qp] is finite.

1.2.7 Residually separable extensions. The inequality 1.2.6.2 is also an equality if L/K is a finite residually separable extension of nonarchimedean fields. This condition is automatic if the residue field of K is perfect, and in particular if K is a local field.

1.2.7.1 Theorem Suppose K is a discretely valued nonarchimedean field, L/K is a finite residually separable extension and F is the maximal unramified extension of K in L. Then L/F is totally ramified, [F : K] = fL/K and eL/K = [L : F ]. In particular [L : K] = eL/K fL/K (1.2.7.1) and L/K is unramified if and only if eL/K = 1. Finally, OL is a free OK -module of rank [L : K].

Proof. By corollary 1.2.4.8 we know that [F : K] = fL/K , and since [F : K] = fF/K as well we deduce that fL/F = 1, i.e. L/F is totally ramiﬁed. By proposition 1.2.5.1, eL/F = [L : F ], and since eF/K = 1 we conclude that eL/K = [L : F ]. The formula 1.2.7.1 then follows from the multiplicativity of the degree. We know that OL is a free OF -module of rank [L : F ] by proposition 1.2.5.1 while OF is a free OK -module of rank [F : K] by part (ii) of theorem 1.2.4.2. It follows that OL is a free OK -module of rank [L : K]. We can improve the last assertion of theorem 1.2.7.1.

1.2.7.2 Corollary If L/K satisifies the condition of the theorem, there is an x ∈ OL such that OL = OK [x] and L = K(x). Proof. Let F be the maximal unramified extension of K in L. We have seen that F = K(y) and OF = OK [y] for any y ∈ OF reducing to a generator y¯ of the residual extension kF /k. Furthermore if τ is any uniformizer of OL then i j OL = OF [τ]. Setting e = eL/K and f = fL/K we see that the ef products y τ for 0 ≤ i < f and 0 ≤ j < e are basis of OL as an OK -algebra. From this it is easily seen that if x is any element of OL with the same reduction as y, then i j the x τ for 0 ≤ i < f and 0 ≤ j < e form a basis of OL as an OK -algebra; by Nakayama it suffices to check that this is so modulo mK . 1.2. EXTENSIONS OF NONARCHIMEDEAN FIELDS 33

Let f be the minimal polynomial of y over K. Since f reduces to the minimal 0 polynomial of y¯, f (y) is a unit in OF . Then if τ is any uniformizer of L,

f(y + τ) = f(y) + f 0(y)τ + bτ 2 = f 0(y)τ + bτ 2 for some b ∈ OL, i.e. f(y + τ) is a uniformizer in OL. If we set x = y + τ then x i j and y have the same reduction modulo mL, and the x f(x) for 0 ≤ i < f and 0 ≤ j < e form an OK -basis of OL. The assertion of the corollary can fail if K is not complete (c.f. exercise 1.2.9.7).

1.2.8 Krasner’s lemma. Fix a nonarchimedean ﬁeld K and an algebraic closure Kalg of K. For x ∈ Kalg we deﬁne

κx = min |xi − xj| (1.2.8.1) i6=j where the xi are the distinct conjugates of x over K.

1.2.8.1 Theorem (Krasner’s lemma) With the above notation, if y ∈ Kalg is such that |x − y| < κx and x is separable over K(y) then x ∈ K(y). Proof. For all s ∈ Gal(Kalg /K(y)) we have

|s(x) − y| = |s(x) − s(y)| = |x − y| and therefore

|s(x) − x| = |s(x) − y + y − x| ≤ |x − y| < κx.

Since s(x) and x are conjugate over K, s(x) = x for all s and consequently x ∈ K(y).

1.2.8.2 Corollary Suppose L/K is a separable algebraic extension of the nonarchimedean ﬁeld K and x, y ∈ L have the same degree over K. If |y − x| < κx then K(y) = K(x).

Proof. Here it is automatic that x is separable over K(y). Since K(x) and K(y) have the same degree over K and K(x) ⊆ K(y), we have K(x) = K(y).

1.2.8.3 Proposition If L/K is a totally and tamely ramiﬁed extension of nonarchimedean ﬁelds of degree e, there is a uniformizer π ∈ OK and a root x ∈ L of Xe − π such that L = K(x).

Proof. Observe ﬁrst that L/K is necessarily separable. If τ (resp. π) is a e × uniformizer of OL (resp. OK ) then τ /π ∈ OL . Since the residual extension is e trivial, we can change π by a unit in OK so as to have τ /π ≡ 1 mod mL, or 34 CHAPTER 1. NONARCHIMEDEAN FIELDS

e e |τ /π − 1| < 1. Set f(X) = X − π, so that |f(τ)| < |π|. If a1, . . . , ae are the roots of f in some algebraic closure of L, the last inequality says that Y |τ − ai| < |π|. (1.2.8.2) 1≤i≤e

e The quantities ai/aj are all roots of X − 1 = 0, and since e is not divisible by the residue characteristic, Xe − 1 is separable modulo m. Then the residue of ai/aj mod m is not 1, so that |ai/aj − 1| = 1 for all i 6= j, i.e. |ai − aj| = |ai| Q e for i 6= j. If |τ − ai| = |ai| for all i then i |τ − ai| = |ai| = |π|, contradicting 1.2.8.2. Therefore there is an i such that |τ − ai| < |ai − aj| for all j 6= i. Since both τ and ai have degree e over K, corollary 1.2.8.2 shows that K(τ) = K(ai) and we may take x = ai. If K is any ﬁeld with a nonarchimedean valuation. For

X n f = anX ∈ K[X] n the Gauss norm of f is

|f|Gauss = max(|a0|,..., |an|). (1.2.8.3)

This is a vector space norm in the sense of section 1.2.2.

1.2.8.4 Proposition Let K be a nonarchimedean field, f ∈ K[X] and let x be a simple root of f in some algebraic closure Kalg of K. For all sufficiently small > 0 there is a δ > 0 satisyfing the following property: for all g ∈ K[X] of the same degree as f such that |f − g|Gauss < δ, there is a unique root y of g such that |y − x| < . For these roots x, y we have K(x) ⊆ K(y). If f is irreducible, so is g, and then K(x) = K(y).

alg Proof. Let x1 = x, x2, . . . , xd be the distinct roots of f in K and replace by a quantity that is smaller than 1, and the |xi − xj| for all i 6= j. Then < κx since the conjugates of x are all to be found among the x1, . . . , xd. Since the expression g(x)/g0(x)2 deﬁnes a continuous function of the coeﬃcients of g, 0 there is a δ > 0 such that |f − g|Gauss < δ implies g (x) 6= 0 and

g(x) < g0(x)2 for all i. By Newton’s method g has root y such that |x − y| < . Since < |x − xi| for all i > 1 we cannot have |xi − y| < if i > 1. This establishes the ﬁrst statement. Since x is a simple root of f it is separable over K, and the containment K(x) ⊆ K(y) follows from Krasner’s lemma. Finally if f is irreducible,

deg(f) = [K(x): K] ≤ [K(y): K] ≤ deg(g) = deg(f) 1.2. EXTENSIONS OF NONARCHIMEDEAN FIELDS 35 and it follows that g is irreducible and that K(x) = K(y). The constant can be made eﬀective, c.f. exercise 2.1.7.7. The hypothesis that x is a simple root of f cannot be dropped. Suppose for example that K is a nonarchimedean ﬁeld of characteristic p > 0 and fa(X) = p X + aX + π where π is a uniformizer in K and a ∈ K. A root of f0 generates an inseparable extension L of K, while for any nonzero a however small, fa has p distinct roots any one of which generates a separable extension L0 of K; in particular L0 6= L.

1.2.8.5 Corollary The completion of an algebraically closed valued ﬁeld is algebraically closed.

Proof. Suppose K is an algebraically closed valued ﬁeld and let Kˆ be the completion of K. Let f ∈ Kˆ [X] be an irreducible polynomial; since K is algebraically closed it is perfect, so f is separable. Let x be a root of f in some ﬁnite extension of Kˆ , choose and δ as in proposition 1.2.8.4. Finally, choose g ∈ K[X] so that |f − g|Gauss < δ. Since f is irreducible, so is g, and g has a root y such that Kˆ (x) = Kˆ (y). Since g ∈ K[X], y is algebraic over K and so y ∈ K. Therefore Kˆ (x) = Kˆ , and x ∈ Kˆ , i.e. f has degree one. ¯ For example Cp, the algebraic closure of Qp, is algebraically closed.

1.2.8.6 Corollary Suppose k is a ﬁeld of characteristic 0 with algebraic closure kalg , and set K = k((T )). The algebraic closure of K is the union of the ﬁelds kalg ((T 1/n)) for all n ≥ 1.

Proof. Any unramified extension of k((T )) has the form `((T )) for some algebraic extension of k. Thus the maximal unramified extension of K is kalg ((T )). Any finite extension of kalg ((T )) is tame, and the result follows.

1.2.8.7 Theorem For any n > 0, a local ﬁeld of characteristic zero has a ﬁnite number of extensions of degree n.

Proof. Let K be a local field of characteristic 0. The residue field field of K has exactly one extension of any given degree, so K has exactly one unramified extension of any given degree. It is therefore enough to show that the number of totally ramified extensions of degree n is finite. We write degree n Eisenstein polynomials in K[X] in the form

n n−1 X + a1X ··· + an−1X + πu

× where ai ∈ m and u ∈ OK , and identify them with points of the space

n−1 × (a1, . . . , an−1, u) ∈ m × OK = X The Gauss norm on K[X] induces the natural topology on X, and in particular X is compact. If f ∈ X, a root of f generates a totally ramiﬁed extension of 36 CHAPTER 1. NONARCHIMEDEAN FIELDS degree n, and this extension is separable since K has characteristic 0. Conversely we have seen that every totally ramiﬁed extension of K arises in this way. We now apply proposition 1.2.8.4 to each f ∈ X with = κx, where x is any root of f in an algebraic extension of K. If δx is the corresponding quantity from the proposition we choose an open ball Bf around f ∈ X of radius δx.

Since X is compact it is covered by finitely many Bf1 ,...,Bfr . If a root of some Eisenstein polynomial f generates a totally ramified extension of K of degree n, f ∈ Bfi for some i, and then K(x) = K(xi) by the proposition; thus K has at most r totally ramified extensions of degree n. We will see in section 2.1.5 that a local field of characteristic p > 0 has infinitely many separable extensions of degree p.

1.2.8.8 Quadratic extensions of Qp. As an example, let’s find all√ quadratic extensions of Qp. By Kummer theory they all have the form Qp( a) and the distinct fields of this type correspond bijectively to the nontrivial elements of × ×2 × ×2 Qp /Qp . So it is enough to work out the structure of Qp /Qp . × × Suppose first p is odd. As groups Qp ' Zp ×Z the second factor is generated × × by a uniformizer. Furthermore Zp ' Fp × U where U is the cyclic subgroup × generated by 1 + p ∈ Zp . Since p does not divide 2, every element of U is a square by the results of section 1.1.6.5. We conclude that

× ×2 × ×2 Qp /Qp ' (Fp /Fp ) × (Z/2Z) where the ﬁrst factor is cyclic of order two, generated by the image of any × u ∈ Zp that is not a square modulo p, and second factor is generated by the class of p. The elements of ×/ ×2 are represented by 1 ,u, p and up, and so Qp Qp √ √ has three quadratic extensions when p is odd, namely ( u), ( p) and Qp √ Qp Qp Qp( up). × ×2 The analysis for p = 2 is similar except for the structure of Q2 /Q2 . By × the results of section 1.1.9.7, x ∈ Z2 has a square root if x ≡ 1 mod 8. On the × other hand (Z2/8Z2) = {1, 3, 5, 7} is the noncyclic group of order 4. It follows × × ×2 that x ∈ Z2 is a square if and only if x ≡ 1 mod 8, and Q2 /Q2 is a product of three√ cyclic groups√ of√ order 2. Thus√ Q2 √has seven√ quadratic extensions,√ namely Q2( 3), Q2( 5), Q2( 7), Q2( 2), Q2( 6), Q2( 10) and Q2( 14).

1.2.9 Exercises

1.2.9.1 (i) Show that any two algebraically closed ﬁelds of the same cardi- ¯ nality are isomorphic. (ii) Deduce from this that C is isomorphic to Qp for any p. In particular, C has a nonarchimedean absolute value extending the p-adic absolute value on Q for any p. (iii) Do you still believe in the axiom of choice? ¯ 1.2.9.2 Show that Qp is not complete for the absolute value extending that of Qp (construct an element of Cp that is not ﬁxed by any open subgroup of ¯ Gal(Qp/Qp)). 1.3. EXISTENCE AND UNIQUENESS THEOREMS. 37

1.2.9.3 A nonarchimedean ﬁeld is spherically complete if any nested sequence B1 ⊇ B2 ⊇ B3 ... of closed balls has nonempty intersection. (i) Show that any discretely valued nonarchimedean ﬁeld is spherically complete. (ii) Show that Cp is not spherically complete.

1.2.9.4 Let K be a nonarchimedean ﬁeld and L/K an algebraic extension. Show that the conclusion of Hensel’s lemma is valid for L. Likewise for Newton’s method.

¯ 2 1.2.9.5 Let βn be a sequence of elements of Q2 such that β0 = 2 and βn+1 = βn for all n ≥ 0. Let K0 = Q2(β1, β2,...) and and let K be the completion of K0 with respect to the unique absolute value of K0 extending the 2-adic valuation 2 of Q2. (i) Show that X − 3 is irreducible in K[X]. (ii) If L = K(α) where α is 2 ¯ a root of X − 3 in Q2, show that [L : K] = 2 but eL/K = fL/K = 1. (use the previous exercise).

1.2.9.6 Show that a compositum of totally ramified extensions is not necessarily totally ramified (consider the extensions in section 1.2.8.8: which are unramified? totally ramified?)

1.2.9.7 Let R = Z(2), the localization of Z at the maximal ideal (2), so that the fraction ﬁeld R is Q. Let K = Q(α) where α is a root of the polynomial f(X) = X3 + X2 − 2X + 8 (c.f. exercise 1.1.11.5). (i) Show that the 2-adic absolute value of Q has exactly 3 inequivalent extensions v0, v1, v2 to K. (ii) Show that the integral closure S of R in K is the set of x ∈ K such that vi(x) ≥ 0 for i = 0, 1, 2. (observe that S is a Dedekind domain). (iii) Show that there is no x ∈ S such that S = R[x] (ﬁrst show that if this were so, the prime ideals of S above (2) would be in a one-to-one correspondence with the distinct irreducible factors of f modulo 2).

1.3 Existence and Uniqueness theorems.

In this section we will determine the structure of equicharacteristic nonarchimedean fields, and show that for any perfect field k of characteristic p there is an absolutely unramified, mixed characteristic nonarchimedean field with residue field k, which is furthermore unique up to canonical isomorphism.

1.3.1 p-rings A p-ring is a pair (A, I) consisting of a commutative ring A with an ideal I such that A is I-adically complete and separated and the quotient ring A/I is a perfect ring of characteristic p > 0. A morphism f : (A, I) → (A0,I0) of p-rings is a ring homomorphism A → A0 sending I into I0. If (A, I) → (A0,I0) is a morphism of p-rings, the reduction of f is the homomorphism A/I → A0/I0. If (A, I) is a p-ring then p ∈ I and A is p-adically complete and separated. 38 CHAPTER 1. NONARCHIMEDEAN FIELDS

Then p ∈ I implies that

x ≡ y mod In ⇒ xp ≡ yp mod In+1 (1.3.1.1) for all x, y ∈ A and n > 0. A p-ring is strict if I = (p) and p is not a divisor of zero in A. The argument of proposition 1.1.8.1 can be used with almost no change to show that if (A, I) is strict, any element of A has a unique p-adic expansion

X n a = [an]p n≥0 with an ∈ A/I for all n. As before a system of representatives of (A, I) is a section s : A/I → A the canonical projection. A system of representatives s is additive if s(a + b) = s(a) + s(b) and multiplicative if s(ab) = s(a)s(b). If s is both additive and multiplicative, the image of s is a subring of A. We will shows that any p- ring has a multiplicative system of representatives. We will need the following general result whose proof is left as an exercise:

1.3.1.1 Lemma Let A be a commutative ring I ⊂ A an ideal and suppose that A is I-adically complete and separated. Let Un ⊆ A for n > 0 be a sequence of nonempty subsets such that Un+1 ⊆ Un for all n > 0, and if x, y ∈ Un then n+1 T x ≡ y mod I . The intersection n>0 Un is a singleton. As usual we denote the canonical surjection A → A/I by x 7→ x¯. For a ∈ A/I and n > 0 let Un(a) ⊂ A be the subset deﬁned by

n x ∈ Un(a) ⇐⇒ x¯ = a and x is a p th power. (1.3.1.2)

Evidently Un+1(a) ⊆ Un(a) for all n. The next lemma shows that we can apply lemma 1.3.1.1 to the sets Un(a):

1.3.1.2 Lemma For all a ∈ A/I and n > 0, Un(a) is nonempty, and if x, n+1 y ∈ Un(a) then x ≡ y mod I .

n Proof. Since A/I is perfect there is a b ∈ A/I such that bp = a. Then if u ∈ A pn is such that u¯ = b, u ∈ Un(a). pn pn Suppose now x and y are elements of Un(a) and write x = u , y = v . n n Since x¯ =y ¯ = a, u¯p =v ¯p and thus u¯ =v ¯. This says u ≡ v mod I, n n and repeated application of 1.3.1.1 yields up ≡ vp mod In+1, i.e. x ≡ y mod In+1.

1.3.1.3 Proposition Any p-ring (A, I) has a unique system of representatives s : A/I → A which commutes with the pth power map: s(ap) = s(a)p. This s is multiplicative, and x ∈ A is in the image of s if and only if it is a pnth power for all n > 0. If A has characteristic p, s is additive as well. 1.3. EXISTENCE AND UNIQUENESS THEOREMS. 39

Proof. By lemmas 1.3.1.1 and 1.3.1.2 we can define a function s : A/I → A by \ {s(a)} = Un(a). n>0 By construction s(a) reduces to a modulo I, i.e. s is a system of representatives. From the definition 1.3.1.2 we see that the pth power map sends Un(a) into p p p Un+1(a ), from which it follows that s(a) = s(a ). We next show that x ∈ A is in the image of s if and only if it is a pnth power for all n > 0. Since A/I is perfect the condition is clearly necessary; on n the other hand if x is a p th power for all n > 0 and a =x ¯ then x ∈ Un(a) for all n and thus x = s(a). From this it follows that s : A/I → A is the unique system of representatives such that s(ap) = s(a)p: if s0 is another such map then s0(a) is a pnth power for all n, hence s0(a) = s(b) for some b ∈ A/I. Reducing modulo I yields a = b and then s0(a) = s(a). For any a, b ∈ A/I the elements s(ab) and s(a)s(b) are pnth powers for all n, and have the same reduction, namely ab. Therefore s(ab) = s(a)s(b) for all a and b in A/I, so s is multiplicative. Suppose finally A has characteristic p. Then

−n n −n n −n −n n s(a) + s(b) = s(ap )p + s(bp )p = (s(ap ) + s(bp ))p is a pnth power for all n, so that s(a)+s(b) = s(c) for some c. Reducing modulo I yields c = a + b and then s(a) + s(b) = s(a + b), so s is additive. The s(a) in the proposition is called the Teichmüller lift of a, and we will denote it by [a]. Teichmuller lifts are functorial in the following sense:

1.3.1.4 Corollary If f :(A, I) → (A0,I0) is a morphism of p-rings with residue ﬁelds k, k0, the diagram [] k / A

f¯ f [] k0 / A0 commutes. Proof. In fact for all x ∈ k, f([x]) is a pnth power for all n > 0 and reduces to f(x) ∈ k0.

1.3.1.5 Remark. The commutative diagram in the corollary says that f([a]) = [f(a)], so if (A, I) is a strict p-ring the morphism f is

X n X ¯ n f( [an]p ) = [f(an)]p . (1.3.1.3) n≥0 n≥0

In particular, f is the unique morphism lifting f¯. 40 CHAPTER 1. NONARCHIMEDEAN FIELDS

1.3.2 Equicharacteristic nonarchimedean fields. As a first application we show that any complete equicharacteristic nonarchimedean field is a field of Laurent series.

1.3.2.1 Theorem Suppose K is a discretely valued equicharacteristic nonarchimedean field. If the residue field of K is perfect, K has a field of representatives, and there is an isomorphism K ' k((T )) for which the splitting s : k → OK is the identity on k.

Proof. We first show that K has a field of representatives. When K and k have positive characteristic this follows from proposition 1.3.1.3, so we suppose K and k have characteristic zero. Then Q maps injectively to K and k, and thus into OK . In particular any subfield of OK must map injectively to k. By Zorn’s lemma there is a maximal subfield F ⊆ OK , and we denote by F¯ its image in k, so that F → F¯ is an isomorphism. We claim that k is algebraic over F¯. If for example t ∈ k is transcendental over F¯, it is the image under reduction of some x ∈ OK which must itself be transcendental over F since F → F¯ is an isomorphism. Then F (x) is a a proper extension field of F , contradicting the maximality of F . Suppose now t ∈ k and f¯(X) ∈ F¯[X] is the minimal polynomial of t. Since F¯ has characteristic zero, f¯ is separable. If f ∈ F [X] is monic and reduces to f¯, it is irreducible and has a root x ∈ OK reducing to t by the corollary to Hensel’s lemma. Since F is a maximal subfield of OK , F (x) = F , whence F¯(t) = F¯ and t ∈ F¯. We conclude that F¯ = k and F is a field of representatives. If π is a uniformizer of K, any element of OK has a unique series represen- P k tation of the form k s(ak)π . It follows that the map f : k[[T ]] → OK defined by X k X k akT 7→ s(ak)π k k is an isomorphism of rings, and passing to the quotient field yields a isomorphism k((T )) → K of nonarchimedean fields. If K has characteristic p > 0 and k is not perfect it is still true, although much more difficult to prove, that K ' k((T )). In fact there is no canonical isomorphism of this sort, or canonical identification of k with a subfield of OK .

1.3.3 The universal p-ring on a set. Let S be a set and Z[S] the polynomial ring on the elements of S. For n ∈ N we set Rn = Z[S] and deﬁne

−∞ [Sp ] = lim R Z −→ n n

p where Rn → Rn+1 is the homomorphism induced by X 7→ X for all X ∈ S. We −∞ −n can think of Z[Sp ] as the polynomial ring in the variables Xp for all n ≥ 0 p−n p−∞ and X ∈ S; X can be identiﬁed with the image if X ∈ S ⊂ Rn in Z[S ]. p−∞ p−∞ We denote by iS : S → Z[S ] the composite map S → Z[S] → Z[S ] where 1.3. EXISTENCE AND UNIQUENESS THEOREMS. 41

−∞ the ﬁrst map is the canonical one for a polynomial ring, and Z[S] → Z[Sp ] uses the identiﬁcation Z[S] = R0. Since inductive limits commute with quotients

p−∞ p−∞ p−∞ Fp[S ] = Z[S ]/pZ[S ] (1.3.3.1) ¯ ¯ is the inductive limit of the polynomial rings Rn = Fp[S] where again Rn → p R¯n+1 is induced by X 7→ X for X ∈ S. Since the pth power homomorphism is ¯ ¯ the identity on Fp, Rn → Rn+1 is the pth power homomorphism on Fp[S], and p−∞ it follows that Fp[S ] is the perfection of Fp[S]. p−∞ p−∞ Finally, we deﬁne Zp{S } to be the p-adic completion of Z[S ]. Since p−∞ p−∞ p is not a zero-divisor in Zp[S ] it is not a zero-divisor in Zp{S }. Since

p−∞ p−∞ p−∞ Fp[S ] = Z{S }/pZ{S }.

−∞ it follows that Z{Sp } is a strict p-ring. p−∞ p−∞ The map iS induces maps S → Fp[S ], S → Z{S } which we also denote by iS, as they are all compatible in the obvious sense.

1.3.3.1 Proposition Let S be a set.

(i). Let A be a perfect ring of characteristic p. Any map of sets f : S → A of [ p−∞ S extends uniquely to a homomorphism f : Fp[S ] → A such that

iS p−∞ S / Fp[S ] (1.3.3.2)

f [ f # A

commutes.

(ii). Let (A, I) be a p-ring. Any map of sets f : S → A/I extends uniquely to ] p−∞ ] [ a homomorphism f : Zp{S } → A such that f = f .

p−∞ Proof. If A is perfect of characteristic p we set Rn = Fp[S] and identify Fp[S ] with lim R . Then f ] is homomorphism induced by the homomorphisms send- −→n n p−n ] ing X ∈ S ⊂ Rn to f(X) . The uniqueness of f is a general property of inductive limits. −∞ If (A, I) is a p-ring we set R = [S] and identify [Sp ] with lim R . n Z Z −→n n p−n The homomorphisms Rn → A which send X ∈ S ⊂ Rn to [f(X)] yield −∞ a homomorphism Z[Sp ] → A whose reduction is the one constructed pre- −∞ viously. Since p ∈ I the completion of Z[Sp ] → A is a homomorphism ] p−∞ ] f : Zp{S } → A. The uniqueness of the lifting f follows from remark 1.3.1.5. 42 CHAPTER 1. NONARCHIMEDEAN FIELDS

p−∞ p−∞ 1.3.3.2 Corollary The constructions Fp[S ], Zp{S } are functorial in S.

Proof. This is easily checked directly, but in any case the functoriality of p−∞ p−∞ Fp[S ] is a general property of free objects, and that of Zp{S } follows from its universal property.

1.3.3.3 Corollary If g : A → B is a homomorphism of perfect rings of characteristic p and f : S → A is a map of sets then (gf)[ = gf [.

Proof. The uniqueness part of proposition 1.3.3.1 shows that the right hand diagonal arrow in the commutative diagram

iS p−∞ S / Fp[S ] gf [ f [ f # g # A / B is equal to (gf)[.

p−∞ The ﬁrst part of proposition 1.3.3.1 asserts that Fp[S ] and the map p−∞ iS : S → Fp[S ] is a free object in the category of perfect rings. We will p−∞ p−∞ accordingly call Fp[S ] the free perfect ring on S. The ring Zp{S } is not a free object in this precisely this sense and we will call it instead the universal p-ring on S. p−∞ p−n Since elements of Fp[S ] may be thought of as polynomials in the X there is an obvious sense in which one can substitute elements of a perfect ring A for the various X ∈ S appearing in f. This is precisely what the homomorphism f ] in part (i) of proposition 1.3.3.1 does: f : S → A can viewed as an assignment p−∞ ] of values to the various X ∈ S; then for any g ∈ Fp[S ], f (g) is the result of substituting these values into g.

1.3.4 The Uniqueness Theorem The next theorem shows that part (ii) of proposition 1.3.3.1 is a consequence of part (i).

1.3.4.1 Theorem Let (A, I) be a strict p-ring and (A0,I0) a p-ring. For any 0 0 homomorphism f0 : A/I → A /I there is a unique morphism of p-rings f : 0 0 ¯ (A, I) → (A ,I ) such that f = f0.

Proof. Uniqueness follows for the formula of equation 1.3.1.3, given that f exists. As for existence we set S = A/I; for a ∈ A/I it will be convenient to denote 0 0 0 the corresponding element of S by Xa. The maps g : S → A/I, g : S → A /I ] 0 ] given by g(Xa) = a, g(Xa) = f(a) induce homomorphisms of p-rings g , (g ) 1.3. EXISTENCE AND UNIQUENESS THEOREMS. 43 sitting in a diagram p−∞ Zp{S } (g0)] g] $ A / A0 and we want to ﬁll in the dotted arrow with a homomorphism. The homo- ] P n P n morphism g is surjective since g maps n≥0[Xan ]p to n≥0[an]p , so it 0 P i ] suﬃces to show that Ker(g) ⊆ Ker(g ). Suppose h = i≥0[hi]p ∈ Ker(g ); ] P [ i ] [ since g (h) = i≥0[g (hi)]p , h ∈ Ker(g ) says that g (hi) = 0 in A/I for 0 0 [ [ all i. Then g = fg implies that (g ) (hi) = f(g (hi)) = 0 for all i, so that h ∈ Ker((g0)]). The uniqueness theorem follows by taking (A0,I0) = (A, I):

1.3.4.2 Corollary Let k be a perfect ring of characteristic p > 0. If A and A0 are strict p-rings with residue ring k there is a unique isomorphism A −→∼ A0 reducing modulo p to the identity of k.

1.3.5 The existence theorem. We now apply the universal p-ring construction to

S = {X0,X1,X2,...Y0,Y1,Y2,...}. (1.3.5.1) If we set X n X n x = [Xn]p , y = [Yn]p n≥0 n≥0 then for any polynomial F ∈ Z[X,Y ],

X F n F (x, y) = [hn (X·,Y·)]p (1.3.5.2) n≥0

F p−∞ for some sequence of hn (X·,Y·) ∈ Fp[S ]. Similar considerations apply to polynomials in any number of variables. F The polynomials hn so deﬁne give a “universal” method of computing F in any strict p-ring (A, I). As we observed earlier, any a ∈ A has a unique series expansion X n a = [an]p (1.3.5.3) n with an ∈ A/I. We can use 1.3.5.3 to identify the a ∈ A with the sequence N (an)n∈N ∈ (A/I) . Thus any strict p-ring has a canonical “coordinate system” in which the coordinates are elements of A/I.

1.3.5.1 Proposition Suppose (A, I) is a p-ring and a = (ai), b = (bi) ∈ A. F F For any F ∈ Z[X,Y ], F (a, b) = (hi (a·, b·)) where the hi are deﬁned by 1.3.5.2. 44 CHAPTER 1. NONARCHIMEDEAN FIELDS

Proof. Apply proposition 1.3.3.1 with S as in 1.3.5.1 and f sends Xi 7→ ai and [ [ [ Yi 7→ bi. Then f ((X·)) = a, f ((Y·)) = b, and since f is a homomorphism [ f (F ((X·), (Y·))) = F (a, b). On the other hand F ((X·), (Y·)) = (hi(X·,Y·)) by [ deﬁnition, so f (F ((X·), (Y·))) = (hi(a·, b·). If we take F (X,Y ) = X +Y , F (X,Y ) = XY or F (X) = −X the proposition p−∞ yields Sn, Pn, Mn ∈ Fp[S ] such that

(ai) + (bi) = (Si(a·, b·)), (ai)(bi) = (Pi(a·, b·)), −(ai) = (Mi(a·) (1.3.5.4) for any strict p-ring A and (ai), (bi) ∈ A. In other words the ring operations in a strict p-ring can be computed by means of “universal” formulas in the coordinates. The commutative and associative laws for addition are expressed by the formulas

Si(X·,Y·) = Si(Y·,X·),Si(X·,S·(Y·,Z·)) = Si(S·(X·,Y·),Z·)) (1.3.5.5) and similarly for the remaining ring axioms. Note also that

p(a0, a1,...) = (0, a0, a1,...) (1.3.5.6) in this notation.

1.3.5.2 Theorem For any perfect ring k of characteristic p > 0 there exists a strict p-ring with residue ring k, which is unique up to unique isomorphism.

Proof. We have already proven uniqueness. For existence we consider the ring −p−∞ [ −p−∞ Zp{S } with S = k, let g : S → k be the identity and g : Fp[k ] → k −p−∞ is the universal homomorphism. As before we identify elements of Zp{k } −p−∞ −p−∞ with vectors of elements of Fp[k ]. Deﬁne a relation on Zp{k } by

[ [ (ai) ∼ (bi) if and only if g (ai) = g (bi) for all i ≥ 0.

This is evidently an equivalence relation and we deﬁne A to be the quotient −p−∞ of Zp{k } by this relation. By construction we can identify elements of A vectors of elements of k. 0 0 If (ai) ∼ (ai) and (bi) ∼ (bi) then

0 0 0 0 (ai) + (bi) = (Si(a·, b·)), (ai) + (bi) = (Si(a·, b·)) and

[ [ [ [ 0 [ 0 [ 0 0 g (Si(a·, b·)) = Si(g (a·), g (b·)) = Si(g (a·), g (b·)) = g (Si(a·, b·))

0 0 which show that (ai)+(bi) ∼ (ai)+(bi). Similar calculations show that (ai)(bi) ∼ 0 0 0 (ai)(bi) and −(ai) ∼ −(ai). −p−∞ It follows that the ring operations in Zp{k } descend to A. The ring ax- −p−∞ −p−∞ ioms hold for these operations since they hold in Zp{k } and Zp{k } → 1.3. EXISTENCE AND UNIQUENESS THEOREMS. 45

A is surjective. The relation 1.3.5.6 and the above description of A show that A is p-adically separated and complete. We also see from 1.3.5.6 that p is not a divisor of zero in A and that the map sending the class of (an) to a0 identifies A/pA ' k. Thus (A, (p)) is a strict p-ring with residue ring k. We denote the unique strict p-ring with residue ring k by W (k); it is called the ring of Witt vectors of k. Taking k to be a perfect field, we find:

1.3.5.3 Corollary For any perfect field k of characteristic p > 0 there is a complete, absolutely unramified mixed characteristic discrete valuation ring W (k) with residue field k, and a nonarchimedean field K(k) with integer ring W (k). The ring W (k) and the field K(k) are unique up to unique isomorphism. Proof. By construction W (k) is a local ring whose maximal ideal is generated by p, so W (k) is a discrete valuation ring; the other assertions are evident. To be sure, uniqueness must be understood as the assertion that if A is any complete absolutely unramified discrete valuation ring of mixed characterstic p > 0 there is a unique isomorphism W (k) → A reducing to the identity of k.

1.3.5.4 Corollary If A is a complete discrete valuation ring of mixed characteristic p > 0 with perfect residue ﬁeld k, there is a unique injective homomorphism W (k) → A reducing to the identity of k. If e is the ramiﬁcation index of W (k) → A then A is a free W (k)-module of rank e.

1.3.5.5 Corollary Suppose A is a complete discrete valuation ring with perfect residue field k, and that k0/k is a perfect extension of k. There is an extension A → A0 of discrete valuation rings such that A0 is complete and the reduction of A → A0 modulo p is k → k0. Proof. If the fraction field K of A is equicharacteristic p then A ' k[[T ]] and we may take A0 = k0[[T ]]. In the mixed characteristic case we know that A ' W (k)[X]/(f) for some Eisenstein polynomial f. Then f is irreducible and W (k0) and A0 = W (k0)[X]/(f) is a complete discrete valuation ring with residue field k0.

1.3.6 Lifts of Frobenius. If k is a perfect ring we can apply theorem 1.3.4.1 to the Frobenius homomorphism k → k, i.e. the map x 7→ xp. The resulting isomorphism is usually denoted by σ : W (k) → W (k). In terms of vectors, it is

p σ(an) = (an) (1.3.6.1) as is easily checked. We also use σ to denote the obvious (and unique) extension to K(k). Finally, we will use the same notation σ for lifts of the qth power map x 7→ xq for any (ﬁxed) power q of p. If A is any commutative ring and I ⊂ is an ideal such that A/I has characteristic p > 0, a lift of the qth power Frobensius to A is a homomorphism σ : A → A reducing modulo I to the qth power map. 46 CHAPTER 1. NONARCHIMEDEAN FIELDS

1.3.6.1 Proposition Let A be a complete discrete valuation ring of mixed characteristic p > 0 with perfect residue field k. If the fraction field of A is a normal extension of the fraction field of W (k), the map σ : W (k) → W (k) extends to an automorphism of A.

Proof. If K is the fraction field of A, the map σ : K(k) → K(k) extends to an automorphism of K by Galois theory. Since A is the integral closure of W (k) in K, this σ induces an isomorphism of A. Note that no claim is made concerning uniqueness. If K is a nonarchimedean field of equicharacteristic p > 0 with perfect residue field k, the existence of lifts of Frobenius is evident: we know that K ' k((T )), and thus X n X p n σ( anT ) = anT (1.3.6.2) n n is a lift of Frobenius. Note that we can not use the Frobenius of K itself here, since this is not an automorphism of K or OK . Furthermore we will in the sequel need lifts of Frobenius that fix a given uniformizer of K.

1.3.6.2 Lemma Suppose K is a discretely valued nonarchimedean ﬁeld with algebraically closed residue ﬁeld k of characteristic p > 0. If σ : K → K is a lift of the qth power Frobenius, then for any a1, . . . , an ∈ OK there is a v ∈ OK such that σn σn−1 v + a0v + ··· + an−1v + an = 0. (1.3.6.3)

If one of the ai is a unit, v can be taken to be a unit.

n Proof. Let π be a uniformizer of K. Dividing 1.3.6.3 by a suitable power of πσ shows that we can assume one of the ai is a unit. We solve 1.3.6.3 by successive approximations. Modulo π it says that

qn qn−1 v + a0v + ··· + an−1v + an = 0 which can be solved since k is algebraically closed, and the solution is nonzero in k. Suppose now v has been found modulo πk and set

σn σn−1 k v + a0v + ··· + an−1v + an = π b.

If we put v = u + πkw then u will be a solution modulo πk+1 if

σn k pn σn−1 k pn−1 k k (v + π w ) + a0(v + π w ) + ··· + an−1(v + π w) + an = π b modulo πk+1, or in other words if w satisﬁes

pn pn−1 w + a1w + ··· + an−1w + a1 = b modulo π. Since k is algebraically closed this has a solution. 1.3. EXISTENCE AND UNIQUENESS THEOREMS. 47

1.3.6.3 Corollary For any lift σ : K → K of the qth power Frobenius, there is a uniformizer π of OK ﬁxed by σ.

σ × Proof. If π is any uniformizer, π = uπ for some u ∈ OK . By the lemma there × σ −1 is a v ∈ OK such that v = uv, and then v π is ﬁxed by σ.

n 1.3.6.4 Corollary If a ∈ K× the endomorphism x 7→ axσ − x of K is surjective if a has positive valuation, or if a has negative valuation and σ is an isomorphism, or if k is algebraically closed. Proof. If a has positive valuation,

n axσ − x = −y (1.3.6.4) can be solved for x by recursive substitution:

X in x = aiyσ . i≥0 If a has negative valuation and σ is an isomorphism we rewrite 1.3.6.4 as

n −n n −n (axσ ) − a−σ (axσ )σ = −y which is solvable by the previous case. Finally when k is algebraically closed × and a ∈ OK we can rescale x so as to have y ∈ OK , which requires changing a by a unit. Then equation 1.3.6.4 is solvable by lemma 1.3.6.2.

1.3.6.5 Theorem Suppose K is a discretely valued nonarchimedean field with residue field k of characteristic p > 0. If σ is a lift of the qth power Frobenius of k to K, the fixed field Kσ of σ is a local field. Proof. If kalg is the algebraic closure of k then K has an unramified extension K0 with residual extension kalg /k, and σ extends to a lift of Frobenius to K0. It suffices to show that (K0)σ is a local field; in other words we may reduce to the case when k is algebraically closed. In particular we may assume that K has a uniformizer π fixed by σ. Suppose first that K is equicharacteristic p > 0; then K ' k((T )) where T is any uniformizer of K. We may assume T is fixed by σ; since σ is continuous σ this means that K = Fq((T )), which is a local field. Suppose now that K has mixed characteristic p > 0 and let π be a uniformizer fixed by σ. Then OK ' W (k)[π] and the minimial polynomial of π is an Eisenstein polynomial f. If the coefficients of f do not belong to W (Fq) then f − f σ would be a polynomial of degree less than deg(f), of which π is a root since it is fixed by σ. Thus f ∈ W (Fq)[X] and σ σ OK ' (W (k)[X]/(f)) ' W (Fq)[X]/(f) ' W (Fq)[π] and it follows that K is a local field. 48 CHAPTER 1. NONARCHIMEDEAN FIELDS Chapter 2

Ramiﬁcation Theory

2.1 The Diﬀerent and Discriminant

2.1.1 Derivations. Let A be a commutative ring, B a commutative A- algebra and let M be a B-module. An A-derivation of B into M is an additive map D : B → M such that D(A) = 0 and

D(xy) = xD(y) + yD(x) (2.1.1.1) for all x, y ∈ B. If D and D0 are derivations of B into M then so are D + D0 and bD for all b ∈ B, so the derivations form a B-module, which we denote DerA(B). Let D(M) be the B-algebra B ⊕ M with the evident addition and multiplication deﬁned by (b ⊕ m)(b0 ⊕ m0) = bb0 ⊕ bm0 + b0m (the algebra of “dual numbers” on M). It is an augmented B-algebra via the homomorphisms b 7→ b ⊕ 0, b ⊕ m 7→ b and we may identify M with a B- submodule of D(M) such that M 2 = 0 for this ring structure. If D is an A-derivation B → M, the map

v : B → D(M) v(b) = b ⊕ D(b) (2.1.1.2) is an A-algebra homomorphism for the A-algebra structure of D(M) induced by the B-module structure. Denote by HomB/A(B,D(M)) ⊂ HomA(B,D(M)) the set of A-algebra homomorphisms whose composition with the augmentation is the identity. Then for any D ∈ DerA(B,M) the map 2.1.1.2 is in HomB/A(B,D(M)), and we have deﬁned a map

DerA(B,M) → HomB/A(B,D(M)) (2.1.1.3) of sets which is evidently functorial in M.

2.1.1.1 Lemma The map 2.1.1.3 is a bijection.

49 50 CHAPTER 2. RAMIFICATION THEORY

Proof. Any v : B → D(M) in HomB/A(B,D(M)) has the form v(b) = b⊕Dv(b) for some A-linear map Dv : B → M. If a ∈ A then Dv(a) = 0, since the A-algebra structure of B(M) comes from the B-algebra structure which is b 7→ 0 b ⊕ 0. The condition that v is a ring homomorphism implies that Dv(bb ) = 0 0 bDv(b ) + b Dv(b). Thus Dv is an A-derivation, and the map v 7→ Dv is inverse to 2.1.1.3.

2.1.2 Diﬀerentials. There is a “universal” A-derivation of B into a B- module. Denote by I the kernel of the multiplication map B ⊗A B → B. The module of diﬀerentials of B/A is the B-module

1 2 ΩB/A = I/I . (2.1.2.1) P P If i ai ⊗ bi ∈ I then i aibi = 0 and then X X aa ⊗ bi = ai(1 ⊗ bi − bi ⊗ 1) i i

1 and it follows the ΩB/A is generated by elements of the form

dx = 1 ⊗ x − x ⊗ 1 + I2. (2.1.2.2)

Easy calculations show that

d(x + y) = dx + dy, d(xy) = xdy + ydx (2.1.2.3)

1 and da = 0 for any a ∈ A. It follows that d is an A-derivation d : B → ΩB/A. 1 If u : M → ΩB/A is any B-module homomorphism the composition u ◦ d is a A-derivation of B into M. We will show that every A-derivation B → M has 1 this form; this is the sense in which d : B → ΩB/A is the “universal A-derivation” of B into a B-module.

2.1.2.1 Proposition The map

1 HomB(ΩB/A,M) → DerA(B,M) f 7→ f ◦ d (2.1.2.4) is an isomorphism. Proof. It suﬃces to show that the composite of this map with 2.1.1.3 is a bijection. The structure morphism v0 : B → D(M) is the image of the zero 1 map ΩB/A → M. If v : B → D(M) is any element of HomB/A(B,D(M)), v and v0 coincide on A and thus

0 0 α : B ⊗A B → D(M) b ⊗ b 7→ v0(b)v(b ) deﬁnes an A-algebra homomorphism. Since the composition of v and v0 with the augmentation D(M) → B is the identity,

α(1 ⊗ x − x ⊗ 1) = 0 ⊕ D(x) 2.1. THE DIFFERENT AND DISCRIMINANT 51 and thus α(I) ∈ M; furthermore M 2 = 0 implies α(I2) = 0. Therefore α 1 2 induces a B-module map ΩB/A = I/I → M, and one checks easily that this an inverse to 2.1.2.4.

1 2.1.2.2 Corollary If B = A[X1,...,Xn] then ΩB/A is the free B-module on dX1, . . . , dXn. Proof. Homomorphisms B → D(M) are in a bijective correspondence with maps of the set {X1,...,Xn} to D(M), and the homomorphisms in HomB/A(B,D(M)) correspond to those who take their values in M ⊂ D(M). If Xi 7→ mi ∈ M, the 1 corresponding map ΩB/A → M sends dXi → mi, and for any choice of mi ∈ M 1 there is exactly one such map. It follows that ΩB/A is free on the dXi.

1 2.1.2.3 Functoriality. The construction ΩB/A is functorial in the following sense. If A / A0

f B / B0 0 0 is a commutative diagram, the ring homomorphism B⊗A B → B ⊗A0 B induces 1 1 0 a B-linear map ΩB/A → ΩB0/A0 , or equivalently a B -linear map

0 1 1 B ⊗B ΩB/A → ΩB0/A0 . (2.1.2.5) which sends 1 ⊗ db 7→ df(b). An important special case is when A0 = A and S ⊂ B is a multiplicative system and B0 = S−1B; then 2.1.2.5 is an isomorphism

−1 1 ∼ 1 S ΩB/A −→ ΩS−1B/A. (2.1.2.6) If S ⊂ A is a multiplicative system this leads to an isomorphism

−1 1 ∼ 1 S ΩB/A −→ ΩS−1B/S−1A. (2.1.2.7) Suppose now A → B → C are homomorphisms of commutative rings. For any C-module M the sequence

0 → DerB(C,M) → DerA(C,M) → DerA(B,M) is easily seen to be exact. By proposition 2.1.2.1 we can rewrite this as

1 1 1 0 → HomC (ΩC/B,M) → HomC (ΩC/A,M) → HomC (C ⊗B ΩB/A,M).

It is easily checked that this exact sequence is obtained by applying HomC ( ,M) to the exact sequence

1 1 1 C ⊗B ΩB/A → ΩC/A → ΩC/B → 0 (2.1.2.8) 52 CHAPTER 2. RAMIFICATION THEORY and that the maps in 2.1.2.8 are the ones induced by functoriality. Suppose now B → C is surjective ring homomorphism with kernel J. Then 1 DerB(C,M) = 0 for any C-module M and it follows that ΩC/B = 0. On the other hand if M is a C-module and D : B → M is an A-derivation, D(J 2) = 0. When viewed as a B-module, JM = 0 and from this one de- duces that the map J → M induced by D is B-linear and vanishes on J 2 (use the Leibnitz rule in both cases). Therefore D induces a B-linear map 2 DerA(B,M) → HomB(J/J ,M), and the resulting sequence

2 0 → DerA(C,M) → DerA(B,M) → HomB(J/J ,M) is exact. As before this comes from an exact sequence

2 1 1 J/J → C ⊗B ΩB/A → ΩC/A → 0 (2.1.2.9) when B → C is a surjective A-algebra map with kernel J. The map J/J 2 → 1 2 C ⊗B ΩB/A sends f ∈ J/J to 1 ⊗ df (note that this is well-deﬁned).

1 2.1.2.4 Computation of ΩB/A. The exact sequence 2.1.2.9 provides a useful 1 method of computing ΩB/A when B is realized as a quotient of a polynomial 1 ring C = A[X1,...,Xn] by an ideal I = (f1, . . . , fr). In fact ΩC/A is the 1 free C-module on dX1, . . . , dXn and therefore ΩB/A is the quotient of the free B-module with basis 1 ⊗ dX1,..., 1 ⊗ dXn by the submodule generated by 1 ⊗ df1,..., 1 ⊗ dfr. The next lemma is a simple example of this procedure, which we will use repeatedly is the following:

2.1.2.5 Lemma Suppose A is a commutative ring and B = A[X]/(f) where f ∈ A[X] is a monic irreducible polynomial. If x ∈ B is the image of X ∈ A[X] 1 0 in B then ΩB/A ' B/(f (x)). Proof. We use the exact sequence

2 1 1 (f)/(f) → B ⊗A[X] ΩA[X]/A → ΩB/A → 0

1 and recall that ΩA[X]/A is the free A[X]-module on dX. Therefore B ⊗A[X] 1 ΩA[X]/A a free B-module of rank one, and we may write any element of it as 1 b ⊗ dX. Then the image of f ∈ (f) in B ⊗A[X] ΩA[X]/A is

1 ⊗ df = 1 ⊗ f 0(X)dX = f 0(x) ⊗ dX and the assertion follows.

1 2.1.2.6 Proposition Suppose L/K is a ﬁnite extension of ﬁelds. Then ΩL/K = 0 if and only if L/K is separable. 2.1. THE DIFFERENT AND DISCRIMINANT 53

Proof. Suppose L = K[x] is a simple extension and f is the minimal polynomial 1 0 of x. The lemma shows that ΩL/K ' L/(f (x)). 0 1 If L/K is separable it is simple, so that f (x) 6= 0 and ΩL/K = 0. If L/K is purely inseparable of degree p it is also simple, but this time f 0(x) = 0 and 1 ΩL/K has dimension one. Finally if L/K is inseparable of any degree there is an intermediate extension K ⊂ K0 ⊂ L such that L/K0 is purely inseparable of degree p. We can then apply the exact sequence 2.1.2.8 to K → K0 → L and 1 1 1 we see that ΩL/K0 is a quotient of ΩL/K . It follows that ΩL/K 6= 0.

2.1.2.7 Corollary Suppose A is an integral domain and B is a ﬁnite A-algebra that is also an integral domain. Let K and L be the fraction ﬁelds of A and B 1 respectively. The extension L/K is separable if and only if ΩB/A is a torsion B-module.

Proof. Since B is a finite A-algebra, L/K is a finite extension and thus L/K 1 is separable if and only if ΩL/K = 0. To conclude it suffices to observe that 1 1 ΩL/K = L ⊗B ΩB/A.

2.1.2.8 Proposition Suppose A → B is a local homomorphism of discrete valuation rings making B a ﬁnite A-module. The residual extension k → kB of 1 A → B is residually separable if and only if ΩB/A is generated by dπ for some uniformizer π of B. If so, it is generated by dπ for any uniformizer π.

Proof. We ﬁrst remark that Ω1 ' Ω1 . If π ∈ B is any uniformizer, kB /k kB /A kB ' B/πB and the exact sequence 2.1.2.9 with J = (π) can be written

πB/π2B −→d k ⊗ Ω1 → Ω1 → 0. B B B/A kB /k

If A → B is residually separable, Ω1 = 0. Then πB/π2B → k ⊗ Ω1 kB /k B B B/A 1 is surjective, and ΩB/A is generated by dπ by Nakayama. Conversely if π is 1 2 1 a uniformizer and dπ generates ΩB/A, πB/π B → kB ⊗B ΩB/A is surjective. Therefore Ω1 = 0 and A → B is residually separable. kB /k

2.1.3 The Diﬀerent. Suppose A is a commutative ring and B is a commu- 1 tative A-algebra. The diﬀerent of B/A is the annihilator of ΩB/A:

1 DB/A = AnnB(ΩB/A). (2.1.3.1)

2.1.3.1 Proposition Suppose A is an integral domain with fraction field K, B is a finitely generated A-algebra that is also an integral domain, and L is the fraction field of B. If L/K is finite, then L/K is separable if and only if DB/A 6= 0. 54 CHAPTER 2. RAMIFICATION THEORY

1 Proof. Since B is finitely generated, ΩB/A is a finitely generated B-module, 1 say with generators dx1, . . . , dxs. If L/K is separable, ΩB/A is torsion, and the annihilator of a finitely generated torsion module over an integral domain is 1 1 nonzero. If L/K is not separable we have seen that L⊗B ΩB/A 6= 0. If ω ∈ ΩB/A 1 is such that 1 ⊗ ω 6= 0 in L ⊗B ΩB/A then bω = 0 implies b = 0. Consequently DB/A = 0.

2.1.3.2 Proposition Suppose A is a commutative ring and B = A[X]/(f) where f ∈ A[X] is a monic irreducible polynomial. If x ∈ B is the image of 0 X ∈ A[X] in B then DB/A = (f (x)). Proof. This follows from lemma 2.1.2.5 and the deﬁnition.

1 2.1.3.3 Corollary With the hypotheses of the proposition, ΩB/A is a cyclic B-module.

2.1.4 Nonarchimedean Fields. When L/K is a ﬁnite separable extension of nonarchimedean ﬁelds we will write DL/K for DOL/OK . We will improve on the following lemma later:

2.1.4.1 Lemma If L/K is a totally ramiiﬁed extension of discretely valued nonarchimedean ﬁelds then

vL(DL/K ) ≥ eL/K − 1 (2.1.4.1) with equality if and only if L/K is tamely ramiﬁed.

Proof. We know OL = OK [x] where x is a root of some Eisenstein polynomial

e e−1 f(X) = X + a1X + ··· + ae.

Then DL/K is generated by

0 e−1 f (x) = ex + ··· + ae−1.

i−1 Since vL((i − 1)aie ) ≥ e,

0 e−1 vL(f (x)) ≥ vL(ex ) ≥ e − 1 with equality only when vL(e) = 0, i.e. when L/K is tamely ramiﬁed.

2.1.4.2 Theorem For any finite extension L/K of discretely valued nonarchimedean fields the following are equivalent: (i). L/K is unramified. (ii). Ω1 = 0. OL/OK 2.1. THE DIFFERENT AND DISCRIMINANT 55

(iii). DL/K is the unit ideal. Proof. The last two statements are clearly equivalent. We will show that the first two are as well. Suppose first that L/K is unramified. Then L/K is residually separable, i.e. OK → OL is a finite residually separable extension of discrete valuation rings. By proposition 2.1.2.8, Ω1 is generated by dπ for OL/OK any uniformizer π of L. Since L/K is unramified we may take π ∈ K, in which case dπ = 0 and thus Ω1 = 0. OL/OK Suppose conversely that Ω1 = 0 and let `/k be the residual extension. OL/OK Since Ω1 → Ω1 = Ω1 is surjective, Ω1 = 0 and O → O is OL/OK `/OK `/k `/k K L residually separable. Let K0/K be the maximal unramified extension of K in 1 1 1 L. Then Ω 0 is a quotient of Ω = 0, and thus Ω 0 = 0. By the OL/OK OL/OK OL/OK lemma, 0 0 = vL(DL/K0 ) ≥ eL/K0 − 1 = [L : K ] − 1 from which we see that K0 = L and L/K is unramified.

2.1.4.3 Corollary Suppose L/K is a residually separable finite extension of nonarchimedean fields. If K0/K is the maximal unramified extension of K in 1 1 L, ΩL/K0 ' ΩL/K and DL/K0 = DL/K .

Proof. Apply the exact sequence 2.1.2.8 to OK → OK0 → OL.

2.1.4.4 Theorem For any residually separable ﬁnite extension of discretely valued nonarchimedean ﬁelds L/K,

vL(DL/K ) ≥ eL/K − 1 (2.1.4.2) with equality if and only if L/K is tamely ramiﬁed.

Proof. This follows from corollary 2.1.4.3 and lemma 2.1.4.1. We include the following for the sake of tradition; it will not really be used.

2.1.4.5 Proposition Suppose E/L and L/K is are ﬁnite, residually separable extensions of nonarchimedean ﬁelds. Then

DE/K = DE/LDL/K . (2.1.4.3)

Proof. We can assume E/K is separable since otherwise both sides are 0. Since 1 ΩE/K is a cyclic OE-module it suﬃces to show that

v (Ω1 ) = v (Ω1 ) + v (O ⊗ Ω1 ). (2.1.4.4) E OE /OK E OE /OL E E OL OL/OK

Find x ∈ OL such that OL = OK [x] and y ∈ OE such that OE = OL[y]. Let f (resp. g) be the minimal polynomial of x over K (resp. y over L). There is a polynomial h ∈ K[X,Y ] such that h(x, Y ) = g(Y ). Then OE ' 56 CHAPTER 2. RAMIFICATION THEORY

OK [X,Y ]/(f(X), h(X,Y )), and applying the method of section 2.1.2.4 shows that Ω1 is the quotient of the free O -module with basis 1⊗dX, 1⊗dY by OE /OK E 0 the submodule generated by 1 ⊗ df = f (x) ⊗ dX and 1 ⊗ dh = hx(x, y) ⊗ dX + 0 0 hy(x, y) ⊗ dY . Since hy(x, Y ) = g (Y ), 1 ⊗ dh = hx(x, y) ⊗ dX + g (y) ⊗ dY and the length of Ω1 is v (f 0(x)g0(y)), which is the right hand side of OE /OK E 2.1.4.4.

2.1.4.6 Corollary With the notation of the proposition, the sequence

0 → O ⊗ Ω1 → Ω1 → Ω1 → 0 E OL OL/OK OE /OK OE /OL is exact.

Proof. This is the exact sequence 2.1.2.8 except for the assertion of exactness at the ﬁrst term. However if O ⊗ Ω1 → Ω1 is not injective, the E OL OL/OK OE /OK relation 2.1.4.4 would not hold. In fact the exactness of this sequence is not evident from the proof of proposition 2.1.4.5. The calculation of Ω1 in the proof shows that the injectivity OE /OK of O ⊗ Ω1 → Ω1 is equivalent to h (x, y) being a unit. E OL OL/OK OE /OK x

2.1.5 The Discriminant. If K is any valued field we denote by IK the group of fractional ideals of K, i.e. OK -submodules of K that are free of rank one. When K is discretely valued, IK ' Z, the isomorphism being given by (x) 7→ vK (x). When L/K is a finite extension of valued fields we extend the × × norm map NL/K : L → K by

I = (x) ⇒ NL/K (I) = (NL/K (x)). (2.1.5.1)

This is well-defined since the norm of a unit is a unit. Like the usual norm, the norm on fractional ideals is transitive for successive extensions. If L/K is a finite separable extension of discretely valued fields the discriminant of L/K is the ideal

dL/K = NL/K (DL/K ). (2.1.5.2)

If L/K and E/L are both ﬁnite separable extensions of discretely valued ﬁelds, the transitivity formula 2.1.4.3 and the transitivity of the norm imply that

[E:L] dE/K = dL/K dE/L. (2.1.5.3)

2.1.5.1 Example: cyclotomic ﬁelds. Recall the situation of Example 1.2.5.3: Qp(µpn )/Qp is the extension generated by a root ζ of the polynomial

n Xp − 1 f(X) = Xpn−1 − 1 2.1. THE DIFFERENT AND DISCRIMINANT 57

n−1 and η = ζp . Since

n pn−1 pn p X X − 1 n−1 f 0(X) = − pn−1Xp −1 Xpn−1 − 1 (Xpn−1 − 1)2 the diﬀerent D is generated by pn/(η−1). If v is the normalized valuation Qp(ζ)/Qp n−1 n−1 of Qp(µpn )/Qp, v(p) = (p − 1)p and v(η − 1) = p , so that

v(D ) = (np − n − 1)pn−1. (2.1.5.4) Qp(ζ)/Qp From theorem 2.1.4.2 we deduce:

2.1.5.2 Proposition A finite separable extension L/K of discretely valued nonarchimedean fields is unramified if and only if dL/K = (1).

Suppose L = K(x) for some x ∈ OL such that OL = OK [x]. If f is the 0 minimal polynomial of x over K then DL/K = (f (x)) and therefore fdL/K = 0 alg (NL/K (f (x))). If x = x1, . . . , xn are the conjugates of x in K then f(X) = Q 0 Q i(X − xi), and we ﬁnd f (x) = i>1(x1 − xi), and

Y 2 dL/K = (D) D = (xi − xj) . (2.1.5.5) i6=j

The expression defining D is a symmetric polynomial in the xi and can therefore be written as a polynomial in the coefficients of f. The discriminant dL/K is an isomorphism invariant of the extension L/K. Using it we can show, for example that any local field K of characteristic p > 0 has infinitely many nonisomorphic separable extensions of degree p. It suffices to recall the Eisenstein polynomial

p fa(X) = X + aX + π (2.1.5.6)

0 where π is a uniformizer of K. If L = K(x) where x is a root of fa, fa(x) = a p and thus dL/K = (a ). Letting a → 0 in K produces an infinite series of separable extensions of degree p with distinct discriminants, and thus mutually nonisomorphic. Note that the “limiting extension” in which a = 0 is inseparable. This example shows how the proof of theorem 1.2.8.7 breaks down for a local field of characteristic p > 0. In the notation of the proof take n = p; then all points of X of the form (0,..., 0, u) correspond to inseparable polynomials. They have to be removed in order to use proposition 1.2.8.4, but this results in a noncompact set. Nonetheless the example suggests that the number of finite separable extensions of given degree will be finite if one bounds the discriminant away from zero, and in fact this is true:

2.1.5.3 Theorem Let K be a local ﬁeld of characteristic p > 0. For all positive integers n, m there are only ﬁnitely many separable extensions L/K such that [L : K] = n and v(dL/K ) ≤ m. 58 CHAPTER 2. RAMIFICATION THEORY

Proof. We reuse the setup and notation of the proof if theorem 1.2.8.7. Since the quantity D in 2.1.5.5 is a polynomial function of the coeﬃcients of f, the subset Y ⊂ X deﬁned by v(D) ≤ m is closed, and therefore compact. Since every polynomial corresponding to a point of Y is separable, the argument of theorem 1.2.8.7 can therefore be applied to Y , and the result follows.

2.1.6 The Trace form. The different is usually defined in terms of the trace form of the extension L/K. In this section we compare this definition with the one given above. We first recall part of the proof that a finite extension L/K of fields is separable if and only if the trace TrL/K : L → K is not identically zero. When L/K is separable, say of degree n we may write L = K(x). Then if f is the minimal polynomial of x, ( 0 0 ≤ i < n − 1 Tr (xi/f 0(x)) = . (2.1.6.1) L/K 1 i = n − 1

This equality is most easily seen from the partial fraction expansion

1 X 1 = 0 (2.1.6.2) f(T ) f (xi)(T − xi) 1≤i≤n in which x1, . . . , xn are the roots of f. If both sides are expanded in powers of 1/T , the left hand side starts with 1/T n, while on the right hand side the i i−1 0 coefficient of 1/T is TrL/K (x /f (x)). The formula 2.1.6.1 implies that the trace form hx, yi = TrL/K (xy) is nondegenerate; we will repeat the argument below. Suppose now L/K is a finite separable extension of nonarchimedean fields, ∨ and choose x ∈ OL such that OL = OK [x]. Denote by OL the set of y ∈ L such that TrL/K (xy) ∈ OK for all x ∈ OL; since TrL/K is nondegenerate this is a fractional ideal of L, and thus free OK -submodule of L of rank n = [L : K]. In i 0 fact if we put yi = x /f (x) for all i ≥ 0, 2.1.6.1 implies first that TrL/K (xiyj) ∈ OK for all i, j ≥ 0, and second that

0 0 ··· 0 1 0 0 ··· 1 ∗   Tr (x y ) = . . . . L/K i j 0≤i,jmatrix has ∨ 0 determinant ±1 the yi for 0 ≤ i < n are a basis of OL. Since DL/K = (f (x)), we have shown that ∨ −1 OL = DL/K . (2.1.6.3)

If a is any fractional ideal of L, the K-linearity of TrL/K shows that TrL/K (a) is a fractional ideal of K. 2.1. THE DIFFERENT AND DISCRIMINANT 59

2.1.6.1 Proposition For any fractional ideals a ⊆ OL, b ⊆ OK , TrL/K (a) ⊆ b −1 if and only if a ⊆ bDL/K .

Proof. By 2.1.6.3 and the K-linearity of TrL/K ,

−1 TrL/K (a) ⊆ b ⇐⇒ b TrL/K (a) ⊆ OK −1 ⇐⇒ TrL/K (b a) ⊆ OK −1 ∨ −1 ⇐⇒ b a ⊆ OK = DL/K −1 ⇐⇒ a ⊆ bDL/K .

2.1.7 Exercises.

2.1.7.1 Suppose L/K is a finite separable extension. (i) Show that L ⊗K L is isomorphic to a direct sum of fields. (ii) Use this to give another proof that 1 ΩL/K = 0 when L/K is finite and separable.

2.1.7.2 Suppose L/K is a ﬁnite purely inseparable extension. (i) Show that the kernel of the multiplication map L⊗K L → L is a nonzero nilpotent maximal ideal of L ⊗K L. In particular L ⊗K L is an Artin local ring. (ii) Deduce from 1 1 this that ΩL/K 6= 0. (iii) Use the last result to show that ΩL/K 6= 0 for any ﬁnite inseparable extension.

2.1.7.3 Suppose A is a commutative ring and {Bi} is a ﬁltered system of commutative A-algebras. Show that if B = lim B then −→i i Ω1 ' lim Ω1 . B/A −→ Bi/A i

1 Deduce from this that if L/K is a separable algebraic extension then ΩL/K = 0.

2.1.7.4 Suppose A is a commutative ring of characteristic p > 0 and denote perf 1 by A the perfection of A. Show that ΩAperf /A = 0. Compare with exercise 2.1.7.2.

2.1.7.5 Suppose K is a field of characteristic p > 0 and L/K is an extension such that Lp ⊆ K. A set S ⊂ L is a p-base of L/K if L is generated as an Q n(x) K-vector space by monomials of the form x∈S x where n(x) = 0 for all but finitely many x ∈ S and 0 ≤ n(x) < p for all x. Show that S is a p-base if 1 and only if the dx for all x ∈ S form a basis of the L-vector space ΩL/K . (use exercise 2.1.7.2 to show sufficiency).

2.1.7.6 Suppose K and L are as in the last problem A subset S ⊂ L is p- Q n(x) independent if the monomials x∈S x are linearly independent over K. (i) 60 CHAPTER 2. RAMIFICATION THEORY

Show that a singleton set S = {x} is p-independent if and only if x 6∈ K. (ii) Show that any p-indepedent set extends to a p-base.

2.1.7.7 Let K be a nonarchimedean ﬁeld and f ∈ OK [X] is irreducible and separable. If x is a root of f in some algebraic closure of K and L = K(x), show that the constant in proposition 1.2.8.4 can be taken to be |D|1/2 where dL/K = (D).

2.2 The Ramiﬁcation Filtration

For the rest of this chapter we assume unless otherwise stated that all fields are nonarchimedean with separable residue field and all field extensions are separable (they are also residually separable by the previous hypothesis.

2.2.1 The Inertia Group. Suppose first that L/K is a finite Galois extension with residual extension kL/k. Then kL/k is a Galois extension too, since alg alg a k-embedding of kL into k must lift to a K-embedding of L into K by theorem 1.2.4.2. But any two K-embeddings L → Kalg have the same image, alg so the same must be true for any two k-embeddings of kL into k . By corollary 1.2.4.4 any element of Gal(kL/k) lifts to a unique element of Gal(L/K), or in other words the natural homomorphism Gal(L/K) → Gal(kL/k) is an isomorphism. If L/K is any finite Galois extension, the maximal unramified extension F of K in L is stable under the action of Gal(L/k) since F is the unique unramified extension of K in L with residue field kL. The argument of the last paragraph applies to F/K, resulting in an isomorphism Gal(F/K) → Gal(kL/k). By Galois theory this produces a surjective homomorphism Gal(L/K) → Gal(kL/k) whose kernel is the inertia group IL/K of the extension L/K. Finally if L/K is any Galois extension of K (not necessarily finite) L is the union of its subfields L0/K that are finite and Galois over K, and then Gal(L/K) = lim Gal(L0/K) where L0 runs over the set of such subfields. ←−L0 One can then show that the inverse limit Gal(L/K) → Gal(kL/k) of the ho- 0 momorphisms Gal(L /K) → Gal(kL0 /k) is still surjective and its kernel IL/K sep is the projective limit of the IL0/K . In the special case L = K we write IKsep /K = IK , so that there is an exact sequence

sep alg 0 → IK → Gal(K /K) → Gal(k /k) → 0 (2.2.1.1) that is the projective limit of the exact sequences

0 → IL/K → Gal(L/K) → Gal(kL/k) → 0 (2.2.1.2) for all ﬁnite Galois L/K.

2.2.2 The ramification groups. When L/K is a finite Galois extension the inertia subgroup is the first term in a series of subgroups of G = Gal(L/K), 2.2. THE RAMIFICATION FILTRATION 61

the ramification filtration. Since the action of G stabilizes OL and mL it sta- i n+1 bilizes the mL for all i ≥ 0. Thus G acts on OL/mL for all n ≥ 0 and we define n+1 Gn = Ker(G → Aut(OL/mL ). (2.2.2.1)

The inertia subgroup is G0, and the Gn are all normal subgroups of G. They are variously called the ramification groups or higher inertia groups. The integers n such that Gn 6= Gn+1 are the breaks of the ramification filtration. Evidently s ∈ Gn if and only if vL(s(a) − a) ≥ n + 1 for all a ∈ OK . Since L/K is residually separable we have OL = OK [x] for some x ∈ OL. It is then clear that s ∈ Gn if and only if vL(s(x) − x) ≥ n + 1. We are then led to introduce the function iG(s) = vL(s(x) − x) (2.2.2.2) where x ∈ OL is any element such that OL = OK [x]. Note that iG(1) = ∞ by definition. It is clear from the above that the right hand side of 2.2.2.2 is independent of the choice of x satisfying this condition, and in fact that

s ∈ Gn ⇐⇒ iG(s) ≥ n + 1. (2.2.2.3)

From this we see that

Gn = 1 for any n ≥ max iG(s). (2.2.2.4) s6=1

Since t(x) may be used in place of x for any t ∈ G,

−1 iG(tst ) = iG(s) (2.2.2.5) for all s, t ∈ G. Applying the nonarchimedean triangle equality to st(x) − x = (st(x) − s(x)) + (s(x) − x) shows that

iG(st) ≥ min(iG(s), iG(t)) with equality if iG(s) 6= iG(t). (2.2.2.6)

If H ⊆ G is a subgroup corresponding to the ﬁeld F ⊆ L then OL = OF [x], and it follows that

Hn = H ∩ Gn and iH (s) = iG(s). (2.2.2.7)

The ramiﬁcation ﬁltration of a quotient of G is more problematic and will be dealt with later.

2.2.3 The different. We saw earlier that that vL(DL/K ) is bounded below by eL/K − 1. We can get an exact formula for vL(DL/K ) using the ramification filtration. Recall first that if OL = OK [x] and f is the minimal polynomial of 0 x then DL/K = (f (x)). Now the conjugates of x over K are the s(x) for all Q s ∈ G, so f(X) = s∈G(X − s(x)) and then Y f 0(x) = (x − s(x)). s6=1 62 CHAPTER 2. RAMIFICATION THEORY

It follows that X vL(DL/K ) = iG(s). (2.2.3.1) s6=1

Now iG(s) has the value n on the set Gn−1 \ Gn, so the sum on the right hand side of 2.2.3.1 is X X n(|Gn−1| − |Gn|) = n(rn−1 − rn) n≥0 n≥0 if we set rn = |Gn| − 1 (note that almost all terms of the sum are zero). But P the sum on the right hand side above collapses to n≥0 rn, so X vL(DL/K ) = (|Gn| − 1). (2.2.3.2) n≥0

Since the inertia group is G0, |G0| = eL/K and thus 2.2.3.2 strengthens theorem 2.1.4.4 in the present case.

2.2.3.1 Example: cyclotomic fields again. Picking up from examples 1.2.5.3 n × and 2.1.5.1, we first identify G = Gal(Qp(µpn )/Qp with (Z/p Z) , where the n × a a ∈ (Z/p Z) acts as ζ 7→ ζ . If v is the normalized valuation on Qp(µpn ) then a a−1 iG(a) = v(ζ − ζ) = v(ζ − 1). Denote by U i ⊆ (Z/pnZ)× the set of u ∈ (Z/pnZ)× congruent to 1 modulo pi. If a ∈ U i \ U i+1, ζa−1 is a primitive pn−ith root of 1, and 1 v (ζa−1 − 1) = p (p − 1)pn−i−1 where vp is the extension to Qp(µpn )of the normalized valuation of Qp. Since n−1 i v(x) = (p − 1)p vp(x) for any x ∈ Qp(µpn ), iG(a) = p and the ramification filtration of G is

G0 = G 1 G1 = ··· = Gp−1 = U 2 Gp = ··· = Gp2−1 = U (2.2.3.3) . . n Gpn−1 = U = 1. Since |U i| = pn−i the right hand side of equation 2.2.3.2 is X (p − 1)pn−1 − 1 + (pi − pi−1)(pn−i − 1) 0

2.2.4 The quotients Gn/Gn+1. We may apply the equality 2.2.2.7 to the 0 case where H = G0 is the inertia subgroup, and K = F is the maximal unram- iﬁed extension of K in L. Then OL = OF [π] where π is any uniformizer of L. The formula for iG then yields

iG(s) = vL(s(π) − π) = 1 + vL(s(π)/π − 1) and 2.2.2.3 then shows that for s ∈ G0, s(π) s ∈ G ⇐⇒ ≡ 1 mod πn. (2.2.4.1) n π

n × 0 × We are therefore led to study the filtration U ⊂ OL defined by U = OL and n n × U = 1 + (π ) ⊆ OL for n > 0. (2.2.4.2) Evidently 0 1 × U /U ' kL (2.2.4.3) and n n+1 U /U ' kL for n > 0 (2.2.4.4) where the latter isomorphism identifies the class of 1 + πnx with the class of x in kL (the reader to whom this is new should check these assertions!) From 2.2.4.1 we see that for n ≥ 0 there is a map

n n+1 θn : Gn/Gn+1 → U /U (2.2.4.5)

n which to s ∈ Gn assigns the class of s(π)/π in U . This map is independent of 0 0 × the choice of π: in π is another uniformizer, π = uπ with u ∈ OL . Then if s ∈ n+1 n+1 Gn, s(u) ≡ u mod π and therefore s(u)/u ≡ 1 mod π . Consequently s(π0) s(π) s(u) s(π) = ≡ mod πn+1 π0 π u π and the assertion follows.

2.2.4.1 Lemma The map 2.2.4.5 is an injective homomorphism.

Proof. For s, t ∈ Gn, t(π) is a uniformizer and thus st(π) st(π) t(π) s(π) t(π) = ≡ mod πn+1 π t(π) π π π by the previous discussion. Then θn is a homomorphism, and its kernel is clearly trivial.

2.2.4.2 Proposition Suppose K is a discretely valued nonarchimedean field of equicharacteristic 0. If L/K is a finite Galois extension with group G then G1 = 1 and G0/G1 is cyclic. In particular if the residue field is algebraically closed, G is cyclic. 64 CHAPTER 2. RAMIFICATION THEORY

Proof. If the residual extension is kL/k we know that G0/G1 is isomorphic to a × subgroup of kL , and a finite subgroup of the mulitiplicative group of a field is cyclic. For n > 0, Gn/Gn+1 is isomorphic to a subgroup of the additive group k, which has no elements of finite order; therefore Gn/Gn+1 = 1 for all n > 0 and thus G1 = 1. The last statement is clear since G/G0 is the Galois group of the residual extension.

2.2.4.3 Corollary If K = k((T )) is a discretely valued nonarchimedean ﬁeld of equicharacteristic 0 with algebraically closed residue ﬁeld, the algebraic closure of K is the union of the K(T 1/n) for all n > 0.

Proof. The proposition shows that any ﬁnite extension of K is cyclic. By Kummer theory a cyclic extension of degree n is generated by a root of Xn − a, with the distinct extensions corresponding to the nontrivial classes in K×/K×n. Since the residue ﬁeld is algebraically closed, the latter group is cyclic of order n, generated by the class of T ∈ K×.

2.2.4.4 Proposition Suppose K is a discretely valued nonarchimedean field with residue field of characteristic p > 0. If L/K is a finite Galois extension with group G then G0/G1 is cyclic, G1 is a p-group and G0 ' G1 o (G0/G1). × Proof. As in the last proposition G0/G1 is isomorphic to a subgroup of kL , and × in this case a finite subgroup of kL cannot have order divisible by p. Similarly for n > 0, Gn/Gn+1 is isomorphic to a finite subgroup of the additive group kL, and in this case every nonidentity element of kL has order p. Thus Gn/Gn+1 is an elementary p-group for n > 0, and G1 is a p-group. n If e = |G0/G1, p ≡ 1 mod e for some n, and by replacing n by a multiple n we may assume that |G0| divides ep . Choose a y ∈ G0 whose image in G0/G1 n n is a generator. Then x = yp satisfies xe = yep = 1. On the other hand the n image of x in G0/G1 has order e since p is relatively prime to e. Therefore the subgroup generated by x maps isomorphically to G0/G1.

2.2.4.5 Corollary If L/K is a Galois extension of local ﬁelds, the Galois group is solvable.

Proof. We already know that the inertia group G0 is solvable, and G/G0 is isomorphic the the Galois group of the residual extension. Since the latter is an extension of ﬁnite ﬁelds, G/G0 is cyclic.

2.3 Herbrand’s Theorem

Suppose now H ⊆ G is a normal subgroup corresponding to a normal extension F of K in L. Our aim is to relate the ramiﬁcation ﬁltration of G with that of the quotient G/H. 2.3. HERBRAND’S THEOREM 65

2.3.1 The Herbrand function. If u ≥ −1 is real we deﬁne

Gu = Gn n = due (2.3.1.1) where due is the least integer not less than u. For −1 ≤ u ≤ 0 we deﬁne −1 [G0 : Gu] = [Gu : G0] (it equals 1, except for u = −1), so that [G0 : Gu] is piecwise constant function of u for u ≥ −1. We next deﬁne Z u dt ϕL/K (u) = .g (2.3.1.2) 0 [G0 : Gt]

Because of our deﬁnition of [G0 : Gt] for −1 ≤ t ≤ 0 we have ϕL/K (u) = u in that interval. We will write ϕ(u) for ϕL/K (u) if the extension L/K is understood. The following should be clear:

2.3.1.1 Lemma The function ϕL/K (u) is the unique real-valued function on [−1, ∞) such that

(i). ϕL/K (u) is continuous and piecewise linear for all u ≥ −1;

(ii). ϕL/K (u) is diﬀerentiable for all nonintegral u and

0 1 ϕL/K (u) = . [G0 : Gu]

(iii). ϕL/K (0) = 0 The equality ! 1 X ϕL/K (u) = min(iG(s), u + 1) − 1 (2.3.1.3) |G0| s∈G0 follows from the lemma. In fact the right hand side clearly satisﬁes (i) and (iii). If m < u < m + 1, the derivative of the right hand side is the number of s ∈ G0 −1 −1 such that iG(s) ≥ m + 2, divided by |G0|. This is [G0 : Gm+1] = [G0 : Gu] , so (ii) holds as well.

2.3.2 The upper numbering. Clearly ϕL/K (t) is an increasing function of u and we denote the inverse function by ψL/K (t). The upper numbering of the ramiﬁcation groups is deﬁned by

v ϕ(u) G = Gψ(v) or equivalently G = Gu. (2.3.2.1)

It is clear that ψL/K is characterized by the following properties: it is continuous, piecewise linear, ψL/K (0) = 0, and ψL/K is diﬀerentiable in any interval of the form (ϕL/K (n), ϕL/K (n + 1)), the derivative being [G0 : Gn]. From this we get the formula Z u 0 v ψL/K (u) = [G : G ]dv (2.3.2.2) 0 66 CHAPTER 2. RAMIFICATION THEORY

(note the upper numbering in the integral). Our goal in this section is an analogue of the formula 2.2.2.7 relative to the quotient map G → G/H, in which the upper numbering replaces the original “lower numbering.”

2.3.2.1 Proposition For all σ ∈ G/H,

1 X i (σ) = i (s). (2.3.2.3) G/H e G L/F s∈σ Proof. (Tate) The sum is over elements of the coset σ ∈ G/H. If σ = 1, both sides are +∞, so we assume σ 6= 1 and pick an s ∈ G mapping to σ. Choose x ∈ OL, y ∈ OF such that OL = OK [x], OF = OK [y]. If we set Y a = s(y) − y, b = (st(x) − x) t∈H then X vL(a) = eL/F iG/H (σ), vL(b) = iG(s) s∈σ and so it suﬃces to show that vL(a) = vL(b), i.e a and b generate the same ideal of OL. The minimal polynomial of x over F is Y f(X) = (X − t(x)) t∈H and thus if we write sf for the result of applying s to the coeﬃcients of f, Y sf(x) = (x − st(x)) = ±b. t∈H

s Since y generates OF , the coeﬃcients of f(X) − f(X) are divisible by s(y) − y. Then sf(x) = sf(x) − f(x) is divisible by s(y) − y, or in other words a divides b. There is a g ∈ OK [X] such that y = g(x). Then x is a root of g(X) − y, which must therefore be divisible by f:

g(X) − y = f(X)h(X).

Apply s to this equation and then substitute x for X; the result is

y − s(y) = sf(x)sh(x) and it follows that b divides a. For σ ∈ G/H we deﬁne

j(σ) = max iG(s). (2.3.2.4) s∈σ 2.3. HERBRAND’S THEOREM 67

If the maximum in 2.3.2.4 is acheived for s ∈ σ the triangle inequality 2.2.2.6 shows that

iG(st) = min(iG(s), iG(t)) = min(j(σ), iH (t)) if iG(s) = max iG(st). t∈H (2.3.2.5) From this we see that

σ ∈ GnH/H ⇐⇒ j(σ) ≥ n + 1 (2.3.2.6) and thus j plays the same role for GnH/H as iG/H does for (G/H)n.

2.3.2.2 Lemma iG/H (σ) − 1 = ϕL/F (j(σ) − 1) (2.3.2.7)

Proof. Let s ∈ G be the element of σ for which iG(s) has the maximum value j(σ), and rewrite the equality 2.3.2.3 as 1 X i (σ) = i (st). (2.3.2.8) G/H e G L/F t∈H By 2.3.2.5 this is the same as 1 X iG/H (σ) = min(j(σ), iH (t)) (2.3.2.9) |H0| t∈H since eL/F = |H0|. The equality 2.3.1.3 applied to L/F and H then yields

iG/H (σ) = ϕL/F (j(σ) − 1) + 1 (2.3.2.10) and we are done.

2.3.2.3 Theorem (Herbrand) If v = ϕL/F (u) then

(G/H)v = GuH/H ' Gu/Hu. (2.3.2.11) Proof. Combining 2.3.2.6 and 2.3.2.7 we see that

σ ∈ GuH/H ⇐⇒ j(σ) ≥ u + 1

⇐⇒ ϕL/F (j(σ) − 1) ≥ ϕL/F (u) = v

⇐⇒ iG/H (σ) − 1 ≥ v

⇐⇒ σ ∈ (G/H)v whence the equality on the left. The isomorphism on the right follows from this and 2.2.2.7.

2.3.2.4 Proposition If H ⊆ G is a normal subgroup corresponding to the Ga- lois extension F/K,

ϕL/K = ϕF/K ◦ ϕL/F , ψL/K = ψL/F ◦ ψF/K . (2.3.2.12) 68 CHAPTER 2. RAMIFICATION THEORY

Proof. If we set v = ϕL/F (u), the derivative of ϕF/K (ϕL/F (u)) is

0 0 |(G/H)v| |Hu| ϕF/K (v)ϕL/F (u) = . eF/K eL/F Herbrand’s theorem shows that this is the same as

|(Gu/Hu)| |Hu| |Gu| 0 = = ϕL/K (u) eF/K eL/F eL/K and the characterization of the ϕ function (lemma 2.3.1.1) then yields the ﬁrst equality in the proposition. The second formula follows from the ﬁrst. The formula GvH/H = (G/H)v (2.3.2.13) follows from Herbrand’s theorem 2.3.2.3 and proposition 2.3.2.4. In fact

v (G/H) = (G/H)ψF/K (v) by deﬁnition, and this is the same as

v GψL/F (ψF/K (v))H/H = GψL/K (v)H = G H/H. The equality 2.3.2.13 shows that the upper numbering extends to a filtration on the Galois group of an infinite Galois extensions. If we denote the Galois group of a finite extension L/K by GL, then for successive Galois extensions v v L/K, E/L the canonical projection GE → GL maps GE → GL surjectively. Thus if L/K is a (possibly infinite) algebraic extension with Galois group G,

G = lim G ←− L L where L runs through the set of finite Galois extensions of K in L, and Gv ⊆ G is defined by Gv = lim Gv . (2.3.2.14) ←− L L This filtration is “left continuous” in the sense that \ Gv = Gw. (2.3.2.15) w 0; they are not necessarily integers, even in the case of a finite extension.

2.3.2.5 Cyclotomic fields yet again. The calculation in section 2.2.3.1 of the ramification filtration for Qp(µpn )/Qp shows that  G u ≤ 0  i i−1 i Gu = U 1 ≤ i ≤ n − 1 and p − 1 < u ≤ p − 1 1 u > pn−1 − 1. 2.3. HERBRAND’S THEOREM 69

The Herbrand function ϕ for Qp(µpn )/Qp is then ϕ(u) = u for u ∈ [−1, 0] u − (pi−1 − 1) = i + for 1 ≤ i ≤ n − 1 and u ∈ (pi−1 − 1, pi − 1] (p − 1)pi−1 u − (pn−1 − 1) = n + for u ≥ pn−1 − 1 (p − 1)pn−1 It follows that the breaks of the ramification filtration in the upper numbering are integral and that Gv = U v for integral 0 ≤ v ≤ n. p∞ pn The field Qp(µ ) is the colimit of the finite extensions Qp(µ ) for n → ∞. The Galois group is G = lim ( /pn )× = × and the previous results show ←−n Z Z Zp v v v × that G = U where now U ⊂ Zp is the subgroup of p-adic integers congruent to 1 modulo pv and v ranges through all natural numbers. The Hasse-Arf theorem states that the breaks of the ramification filtration in the upper numbering are integral for any abelian extension of discretely valued nonarchimedean fields with perfect residue fields. We will see more examples.

2.3.2.6 A Kummer extension. Suppose now K = Qp(µpn ), π is a uniformizer pn ¯ of K and L = K(x) where x is a root of f(X) = X − π in Qp. Since f is an Eisenstein polynomial L/K is a totally ramiﬁed extension of degree pn and x is a uniformizer of L. Since K contains the pnth roots of unity L/K is Galois. The Galois group is isomorphic to G = µpn with α ∈ µpn acting on L via x 7→ αx. We may use x to compute the function iG:

iG(α) = vL(αx − x) = vL(α − 1) + 1. If α − 1 is a primitive pith root of unity, n 2n−i vL(α − 1) = p vK (α − 1) = p and thus 2n−i s ∈ Gm ⇐⇒ p ≥ m. The ramiﬁcation ﬁltration is then n Gu = G for u ∈ [−1, p ] n+i−1 n+i = µpn−i for 0 < i < n, and u ∈ (p , p ] = 1 for u > p2n−1 and the Herbrand function is n ϕL/K (u) = u for u ∈ [−1, p ] u − pn+i = (i + 1)pn − ipn−1 + for 0 < i < n − 1 and u ∈ [pn+i, pn+i+1] pi+1 u − p2n−1 = npn − (n − 1)pn−1 + for u ≥ p2n−1. pn 70 CHAPTER 2. RAMIFICATION THEORY

The breakpoints of the ramiﬁcation ﬁltration for the upper numbering are the integers (i + 1)pn − ipn−1 for 0 < i < n − 1 (as predicted by Hasse-Arf).

2.4 The Norm

× × The main result of this section is that the norm NL/K : L → K is surjective if L/K is a finite extension of discretely valued nonarchimedean fields with algebraically closed residual fields (note that L/K is then totally ramified). This is one of the key points in our treatment of local class field theory.

2.4.1 Unramified extensions. Of course the norm can never be surjective for a nontrivial extension since the image of the induced map ΓL → ΓK has × × index [L : K]. In this section we study the homomorphism OL → OK induced by the norm. n n We denote by UL the groups that were designated U is section 2.2.4: the × n elements OL congruent to 1 modulo π , where π is any uniformizer of L, and n × 0 × 0 × similarly for UK ⊆ OK . By definition UL = OL and UK = OK . 2.4.1.1 Lemma Let f : A → B be a homomorphism of abelian groups, and let An ⊆ A, Bn ⊆ B be filtrations such that f(An) ⊆ Bn for all n ≥ 0 and the natural maps A → lim A/A ,B → lim B/B ←− n ←− n n n are isomorphisms. If the maps An/An+1 → Bn/Bn+1 induced by f are injective (resp. surjective, bijective) then so is f. Proof.

n × Note that the second condition in the lemma applies to the ﬁltrations UK ⊆ OK n × and UL ⊆ OL . n n If x ∈ UL we may write x = 1 + π u with u ∈ OL and π a uniformizer of K; then Y ns n 2n NL/K (x) = (1 + π u) ≡ 1 + π TrL/K (u) mod π . (2.4.1.1) s∈G n n This shows that NL/K (UL ) ⊆ UK , and the norm induces maps n n+1 n n+1 Nn : UL /UL → UK /UK (2.4.1.2) for all n ≥ 0. Evidently N0 is the norm on the residual extensions, and if we ﬁx n n+1 n n+1 a choice of π to identify UK /UK ' k and UL /UL ' kL then Nn for n > 0 is the trace.

2.4.1.2 Proposition Suppose L/K is ﬁnite and unramiﬁed. For any n > 0, n n NL/K induces a surjective map UL → UK , and the canonical projection induces an isomorphism × × ∼ × × OK /NL/K (OL ) −→ k /NkL/k(kL ). (2.4.1.3) 2.4. THE NORM 71

Proof. Since the trace is surjective for any ﬁnite separable extension, 2.4.1.2 is n n surjective for all n > 0 and it follows that the induced map UL → UK is also surjective. In particular this is so for n = 1, and applying the snake lemma to the diagram 1 × × 0 / UL / OL / kL / 0

1 × × 0 / UK / OK / k / 0 yields the isomorphism 2.4.1.3, since all vertical arrows are induced by the appropriate norm.

× × 2.4.1.3 Corollary The norm map OL → OK is surjective if and only if the × × residual norm map kL → k is surjective.

0 0 × 2.4.1.4 Lemma If k /k is an extension of finite fields, the norm Nk0/k :(k ) → k× is surjective. Proof. Suppose |k| = q and |k0| = qf . Since the qth power Frobenius generates the Galois group of k0/k, the norm map is x 7→ xn where qf − 1 n = 1 + q + ··· + qf−1 = . q − 1 The kernel of norm has at most n elements, so the image has at least (qf −1)/n = q − 1 elements. Since the image has at most that many elements, Nk0/k is surjective. Combining the lemma with , we find:

2.4.1.5 Proposition Suppose L/K is unramified Galois extension of local fields. × × (i). The norm map OL → OK is surjective. × × (ii). The group K /NL/K L is cyclic of order n, generated by any uniformizer of K. Proof. The first assertion follows from the lemma and proposition 2.4.1.2. For the second, let vK , vL be the normalized valuations of K and L; since L/K is unramified vL is an extension of vK . In the commutative diagram

× × vL 0 / OL / L / Z / 0

N N × × vK 0 / OK / K / Z / 0 the left hand is surjective, and since L/K is unramiﬁed the image of the right hand arrow is nZ, whence the second assertion. 72 CHAPTER 2. RAMIFICATION THEORY

2.4.2 Some totally ramified cases. We now suppose L/K is cyclic of n × prime degree ` and we fix a generator s ∈ G. The filtrations UK ⊂ OK and n × UL ⊂ OL are defined as in the last section. We also choose uniformizers π ∈ OK n n+1 n n+1 and τ ∈ OL and use these to identify UL /UL ' kL and UK /UK ' k for n n n > 0. More precisely, the class of 1 + τ u ∈ UL is identified with the image of n n u in kL, and similarly the class of 1 + π u ∈ UL is identified with the image of u in k. Since G is cyclic of prime order the ramification groups Gn are either G or 1, and we let t be unique break in the filatration, so that Gn = G for n ≤ t and Gn = 1 for n > t. The Herbrand function and its inverse are ( u u ≤ t ϕ (u) = L/K t + `−1(u − t) u ≥ t ( u u ≤ t ψ (u) = L/K t + `(u − t) u ≥ t and the different has valuation

vL(DL/K ) = (t + 1)(` − 1) by formula 2.2.3.2.

n r 2.4.2.1 Lemma For any integer n ≥ 0, TrL/K (mL) = m where

n + m r = r(n) = and m = (t + 1)(` − 1). `

n r n r −1 Proof. By proposition 2.1.6.1 TrL/K (mL) ⊂ m if and only if mL ⊆ m DL/K , i.e. n r −1 n = vL(mL) ≤ vL(m ) + vL(DL/K ) which since eL/K = ` is equivalent to n ≤ `r − m, i.e. (n + m)/` ≤ r. In proposition 2.4.2.3 we will need the values r(ψ(n)) and r(ψ(n)+1). There are two cases. If n < t then ψ(n) = n, and

(t + 1)(` − 1) + n r(n) = . `

The numerator of the fraction on the right is

(t + 1)` − 1 − (t − n) ≥ (n + 2)` − 2 and consequently 2 r(n) ≥ n + 2 − ≥ n + 1. ` 2.4. THE NORM 73

When n ≥ t we have ψ(n) = t + `(n − t) and (t + 1)(` − 1) + t + `(n − t) 1 r(ψ(n)) = = n + 1 − = n ` ` (t + 1)(` − 1) + t + `(n − t) + 1 r(ψ(n) + 1) = = n + 1 ` In short:

n < t ⇒ r(ψ(n)) ≥ n + 1 (2.4.2.1) n ≥ t ⇒ r(ψ(n)) = n and r(ψ(n) + 1) = n + 1. (2.4.2.2)

n 2.4.2.2 Lemma If x ∈ mL then 2n NL/K (1 + x) ≡ 1 + TrL/K (x) + NL/K (x) mod TrL/K (mL ). (2.4.2.3) Proof. We have

Y s X u NL/K (1 + x) = (1 + x) = 1 + x s∈G u where u runs over the set of elements of Z[G] of the form

u = s1 + ··· + sr with s1, . . . , sr distinct elements of G. With this notation r = aug(u), and the terms with aug(u) = 1, ` yield the terms TrL/K (x) and NL/K (x) in 2.4.2.3. We 2n must show that the remaining terms lie in TrL/K (mL ). Observe ﬁrst that if aug(u) 6= ` then su 6= u, for otherwise u = 1+s+···+s`−1 and then aug(u) = `. Since i and ` are relatively prime this implies that siu 6= u for 0 < i < `, and P u therefore aug(u)6=1,` x is a sum of terms of the form

X siu u x = TrL/K ( x). 0≤i<`

u 2n P u Since aug(u) ≥ 2, x ∈ mL and the same is then true for aug(u)6=1, ` x.

2.4.2.3 Proposition Suppose L/K is cyclic and totally ramiﬁed of degree `. ψ(n) n ψ(n)+1 n+1 Then for all n ≥ 0, NL/K (UL ) ⊆ UK and NL/K (UL ) ⊆ UK . There × × are an ∈ k for 0 < n ≤ t and bn ∈ k for n ≥ t such that the induced morphism ψ(n) ψ(n)+1 n n+1 Nn : UL /UL → UK /UK is  ` x n = 0  p anx 0 < n < t Nn(x) = p . (2.4.2.4) atx + btx n = t > 0  bnx n > t 74 CHAPTER 2. RAMIFICATION THEORY

Proof. Since L/K is totally ramiﬁed, NL/K (τ) = aπ where π, τ are the uni- × n formizers chosen above and a ∈ OK ; thus the image an of a in k is nonzero. On the other hand if r is as in lemma 2.4.2.1 the trace induces an OK -linear map n n+1 r r+1 mL/mL → m /m . (2.4.2.5) These are one-dimensional k-vector spaces which our choice of uniformizers τ ∈ OL, π ∈ OK allows us to identify with k. With these identiﬁcations 2.4.2.5 has the form x 7→ bx for some nonzero b ∈ k. The assertions are clear in the case n = 0, and in the remaining cases we must have ` = p. We treat each in turn:

n n+1 0 < n < t: In this case ψ(n) = n, and in this case TrL/K (mL) ⊆ m by 2.4.2.1, so

n n n p n+1 NL/K (1 + τ u) ≡ 1 + NL/K (τ u) ≡ 1 + anπ u mod m

n+1 n+1 by lemma 2.4.2.2. If u ∈ mL this shows NL/K (UL ) ⊆ UK , and in any case p Nn(u) = anu with an 6= 0, as required.

t t n = t > 0: We still have ψ(t) = t, and this time TrL/K (mL) = m and t+1 t+1 TrL/K (mL ) = m by 2.4.2.2. Lemma 2.4.2.2 now yields

t t t t+1 NL/K (1 + τ u) ≡ 1 + TrL/K (τ u) + NL/K (τ u) mod m

t t which shows that NL/K (UL) ⊆ UK , and taking u ∈ mL and again applying t+1 t+1 2.4.2.2 shows that NL/K (UL ) ⊆ UK . Finally the discussion in the ﬁrst p paragraph shows that Nt(u) = btu + atu with nonzero at, bt.

ψ(n) n > t: This time ψ(n) = t + `(n − t) ≥ n + 1 and therefore NL/K (mL ) ⊆ n+1 ψ(n) n 2ψ(n) ψ(n)+1 m . Furthermore TrL/K (mL ) = m and TrL/K (mL ) ⊆ TrL/K (mL ) = mn+1, so that

ψ(n) ψ(n) n+1 NL/K (1 + τ u) ≡ 1 + TrL/K (τ u) mod m .

ψ(n) n ψ(n)+1 n+1 As before NL/K (UL ) ⊆ UK and NL/K (UL ) ⊆ UK , and ﬁnally Nn(u) = bnu with bn 6= 0.

2.4.2.4 Corollary The map Nn is injective for all n except n = t, in which case the sequence

θt t t+1 Nn t t+1 0 → G −→ UL/UL −−→ UL/UL (2.4.2.6) is exact. 2.4. THE NORM 75

Proof. Injectivity for n 6= t is clear from the formulas in the proposition. On the other hand the image of θt is clearly contained in the kernel of Nt: for t = 0 it suﬃces to observe that s(τ)/τ is in the kernel of the norm, and for t > 0 this t t 0 1 is also the case for s(1 + τ )/(1 + τ ). When t = 0 G0 injects into UL/UL and has order ` when t = 0, while the kernel of N0 has order at most `, so θ0(G0) t t+1 must be the exact kernel. Similarly when t > 0, Gt injects into UL/UL , so the p image has order p; on the other hand the kernel is the set of roots of atX +btX , and there are exactly p of these.

2.4.2.5 Theorem Suppose L/K is a finite Galois extension of discretely valued nonarchimedean fields with algebraically closed residue field. Then the norm map × × NL/K : L → K is surjective. Proof. The hypothesis implies that L/K is totally ramified. Then in the commutative diagram

× × vL 0 / OL / L / Z / 0

N N × × vK 0 / OK / K / Z / 0 the right hand vertical arrow is an isomorphism. It thus suffices to show that N : × × OL → OK is surjective. Since G = Gal(L/K) is solvable, the transitivity of the × × norm shows that it suffices to show that N : OL → OK is surjective when L/K is cyclic of prime order. Since the residue field k of K (and L) is algebraically closed, the formulas in proposition 2.4.2.3 show that Nn is surjective for all × × n ≥ 0, and finally lemma 2.4.1.1 shows that N : OL → OK is surjective. 76 CHAPTER 2. RAMIFICATION THEORY Chapter 3

Group Cohomology

In this chapter we review some basic aspects of group cohomology and homology. We assume the reader is familiar with some basic homological algebra. Our sign conventions for the latter are those of Bourbaki.

3.1 Homology and Cohomology

3.1.1 Definitions. Let G be a group. A left G-module is a left module over the integral group ring Z[G] of G. One easily checks that a G-module A can be identified with an abelian group A endowed with a (left) G-action (g, a) 7→ g · a via homomorphisms, i.e. g · (a + b) = g · a + g · b. For example if A is any abelian group we may make it into a G-module by means of the trivial action: g · a = a for all a ∈ A.A morphism of G-modules is just a morphism of left Z[G]-modules. Although right G-modules (i.e. right Z[G]-modules) are less of a concern here, the entire theory described here holds with appropriate changes for right G-modules. In fact there is an exact equivalence of the categories of left and right G-modules: any group is isomorphic to its opposite group by the map i : s 7→ s−1, so the ring Z[G] is likewise isomorphic to its opposite ring. Then a left G-module becomes a right G-module via extension of scalars by i and conversely. If A is a G-module, the cohomology groups of G with coefficients in A are the Ext groups Hn(G, A) = Extn ( ,A) (3.1.1.1) Z[G] Z where Z has the trivial G-action. The standard properties of Ext groups imply corresponding properties for the Hn(G, A). First and foremost, these groups are functorial in A: for any morphism A → B of G-modules there is a canonical homomorphism Hn(G, A) → Hn(G, B) (3.1.1.2)

77 78 CHAPTER 3. GROUP COHOMOLOGY and these are compatible with composition. Second, for any short exact sequence 0 → A → B → C → 0 (3.1.1.3) of G-modules there is a long exact sequence

→ Hn(G, A) → Hn(G, B) → Hn(G, C) −→∂ Hn+1(G, A) → (3.1.1.4) in which the maps Hn(G, A) → Hn(G, B), Hn(G, B) → Hn(G, C) are the ones induced by functoriality (we will recall later how to compute ∂). This long exact sequence 3.1.1.4 is natural in the sense that if

0 / A / B / C / 0 (3.1.1.5)

0 / A0 / B0 / C0 / 0 is commutative with exact rows,

∂ Hn(G, A) / Hn(G, B) / Hn(G, C) / Hn+1(G, A)

∂ Hn(G, A0) / Hn(G, B0) / Hn(G, C0) / Hn+1(G, A0) (3.1.1.6) is commutative. The homology groups of G with coeﬃcients in A are deﬁned by

Z[G] Hn(G, A) = Torn (Z,A) (3.1.1.7) and have similar formal properties: they are functorial in A, and for any exact sequence 3.1.1.3 there is a corresponding long exact sequence

Hn+1(G, C) → Hn(G, A) → Hn(G, B) → Hn(G, C) → (3.1.1.8) which is natural in the sense that a commutative diagram 3.1.1.5 with exact rows gives rise to a commutative diagram analogous to 3.1.1.6. It follows from a general property of the Tor functors that if A is a ﬂat G-module (i.e. ﬂat as a Z[G]-module) then Hn(G, A) = 0 for all n > 0. This is the case in particular when A is projective.

3.1.2 The bar resolution. The groups Hn(G, A) can be computed by taking any resolution of Z by projective Z[G]-modules

d1 d0 · · · → P2 −→ P1 −→ P0 −→ Z → 0. (3.1.2.1) Then n n H (G, A) ' H (Hom(P·,A)) (3.1.2.2) 3.1. HOMOLOGY AND COHOMOLOGY 79

where Hom(P·,A) is the complex

∗ ∗ ∗ d0 d1 d2 HomG(P0,A) −→ HomG(P1,A) −→ HomG(P2,A) −→ . With this notation, 3.1.2.2 becomes

n ∗ ∗ H (G, A) ' Ker(dn)/Im(dn−1). (3.1.2.3)

One has complete freedom to choose P· in making computations, as long as it is a resolution of Z by projective Z[G]-modules. This can be very useful for the study of speciﬁc groups, as we shall see later. But there is one resolution that works for any group, the so-called bar resolution. In this we take Pn to be the free Z[G]-module with basis Gn: Gn Pn = Bn = Z[G] (3.1.2.4) n The elements of G are usually written as n-tuples (s1, s2, . . . , sn) but we will follow tradition and write this as [s1|s2| · · · |sn]. The maps : B0 → Z and dn : Bn → Bn−1 are deﬁned by ([s]) = 1 X d[s1|s2| · · · |sn+1] = s1[s2| · · · |sn] + [s1| · · · |sisi+1| · · · |sn] (3.1.2.5) 1≤i≤n n+1 + (−1) [s1|s2| · · · |sn].

We can show that this is a resolution of Z by constructing contracting homotopy for 3.1.2.1 when Pn = Bn, i.e. a set of G-module maps

η h0 h1 h2 Z −→ B0 −→ B1 −→ B2 −→ (3.1.2.6) satisfying η = 1,

η + d0h0 = 1, (3.1.2.7)

hn−1dn−1 + dnhn = 1, n ≥ 1. In fact η(1) = [ ], hn(s1[s2| · · · |sn] = [s1|s2| · · · |sn] (3.1.2.8) does the trick. n The elements of C (G, A) = Hom(Bn,A) are called n-cochains and can be identiﬁed with functions a : Gn → A. The subgroups

n ∗ n ∗ Z (G, A) = Ker(dn),B (G, A) = Im(dn−1) of Cn(G, A) are called the subgroups of n-cocycles and n-coboundaries respectively (of course this terminology is used for any cochain complex, but we will restrict its use to this particular case). Then 3.1.2.3 says that Hn(G, A) ' Zn(G, A)/Bn(G, A). (3.1.2.9) 80 CHAPTER 3. GROUP COHOMOLOGY

Two n-cocycles a, a0 are cohomologous if the represent the same element of Hn(G, A), and we will write this as a ∼ a0. Thus a ∼ a0 if and only if a0 = ∗ a + dn−1(b) for some (n − 1)-cochain b. If a(s1, . . . , sn) is an n-cochain the definition 3.1.2.5 shows that ∗ dn(a)(s1, . . . , sn+1) = s1a(s2, . . . , sn) X i + (−1) a(s1, . . . , sisi+1, . . . , sn) (3.1.2.10) 1≤i≤n n+1 + (−1) a(s1, . . . , sn) n and thus the condition that a(s1, . . . , sn) ∈ Z (G, A) is just that the right hand side vanishes. The coboundary map ∂ : Hn(G, C) → Hn+1(G, A) in the long exact sequence 3.1.1.4 is computed as follows. Suppose a short exact sequence 3.1.1.3 is given and c ∈ Hn(G, C). Pick an n-cocycle cˆ ∈ Zn(C,A) representing c and then an n-cochain b ∈ Cn(G, B) mapping to c under the map induced by ∗ B → C. One checks easily that dn(b) is a B-valued (n + 1)-cocycle, and since cˆ ∗ was an n-cocycle, dn(b) maps to zero under B → C. In other words the “values” ∗ ∗ of dn(b) lie in A, and there is a unique n-cocycle aˆ mapping to dn(b) under the map induced by A → B. One then checks that the class a of aˆ in Hn+1(G, A) is independent of the choice of b, and then ∂(c) = a. To compute homology we need a resolution of Z by right G-modules. The most direct approach here is to convert the terms of the bar resolution to right G- modules by means of the functor i in section 3.1.1, in which case the differentials 3.1.2.5 become ([s]) = 1 −1 X d[s1|s2| · · · |sn+1] = [s2| · · · |sn]s1 + [s1| · · · |sisi+1| · · · |sn] 1≤i≤n n+1 + (−1) [s1|s2| · · · |sn]. The change of basis −1 −1 −1 [s1|s2| · · · |sn] 7→ [sn | · · · |s2 |s1 ] yields a more appealing formalism: the differentials are now X d[s1| · · · |sn+1] = [s1| · · · |sn]sn+1 + [sn+1| · · · |si+1si| · · · |sn] 1≤i≤n (3.1.2.11) n+1 + (−1) [sn+1|s2| · · · |s2]. In particular ([ ]) = 1, d[s] = [ ]s − [], d[s|t] = [s]t − [st] + [t]. (3.1.2.12) ˇ The right bar resolution B· → Z is the following resolution of the right G-module Z: the nth term is the free right Z[G]-module Gn, with the same notation for the basis elements as in the (left) bar resolution. The differentials are defined by 3.1.2.11, and the augmentation is ([ ]) = 1. 3.1. HOMOLOGY AND COHOMOLOGY 81

3.1.3 Cohomological degree zero and one. Anyone new to this may well wonder what all this formalism is accomplishing. We now give concrete descriptions of Hn(G, A) for small values of n, namely n = 0, 1 and 2. The case n = 0 is simple: a 0-cochain in A is just an element a ∈ A, and ∗ 0 3.1.2.10 says d0(a)(g) = ga − a; thus a ∈ Z (G, A) if and only if a is fixed by the action of G. If we denote the set of G-fixed elements of A by AG, we have shown that H0(G, A) ' AG. (3.1.3.1) In fact this follows from the definition:

H0(G, A) = Ext0 ( ,A) ' Hom ( ,A) ' AG Z[G] Z Z[G] Z where the last isomorphism identiﬁes f : Z → A with f(1) ∈ AG. We thus see that the functor A 7→ AG is left exact, and its right derived functors are the Hn(G, A). The case n = 1 is a little more interesting. A 1-cochain a(g) in A must satisfy s1a(s2) − a(s1s2) + a(s1) = 0 or a(s1s2) = a(s1) + s1a(s2). (3.1.3.2) Such a map G → A is called a crossed homomorphism.A 1-cochain a(s) is a ∗ coboundary, say a = d0b when a(s) = sb − b. When A is a trivial G-module the coboundaries vanish and a crossed homomorphism is just an ordinary group homomorphism, so

H1(G, A) ' Hom(G, A) when A is a trivial G-module. (3.1.3.3)

3.1.4 Cohomological degree two. The case n = 2 is more interesting still. We will use multiplicative notation for both G and A in this section, and write a 2-cochain a(s, t) as as,t; then the condition that as,t is a 2-cocycle is that

s −1 −1 at,uast,uas,tuat,u = 1 (3.1.4.1)

∗ and as,t. It is a coboundary, say a = d1(b) if

s −1 as,t = btbst bs for some 1-cochain (bs). A map a : G × G → A satisfying 5.2.2.6 is traditionally called a factor system. One should note the particular cases

s a1,1 = as,1, a1,s = a1,1 (3.1.4.2) of 5.2.2.6. In a sense to be described shortly the group H2(G, A) classiﬁes extensions of G by A. To see how, observe ﬁrst that any extension

p 0 → A −→i E −→ G → 1 (3.1.4.3) 82 CHAPTER 3. GROUP COHOMOLOGY gives rise to an action of G-module structure on A, by

ga = gag−1 (3.1.4.4) for g ∈ G and a ∈ A (in what follows we will use multiplicative notation for G and A, even though A is an abelian group). Next, choose a section of p, say s 7→ es for all s ∈ G. Since p(eset) = st,

eset = i(as,t)est (3.1.4.5) for some set of as,t ∈ A. The associative law implies that (as,t) is an A-valued 2- 0 0 cocyle, and if (es) is a different choice of section, es = bses for some 1-cochaine, and the corresponding 2-cocycle is a∂(b). In this way the extension 3.1.4.5 gives rise to a well-defined element of H2(G, A). To explain the sense in which H2(G, A) classifies extensions 3.1.4.5 (for a fixed G-module structure on A) we introduce the category EXT(G, A) whose objects are short exact sequences 3.1.4.3 and whose morphisms are commutative diagrams 0 / A / E / G / 1 . (3.1.4.6)

f 0 / A / E0 / G / 1 We will usually denote the extension 3.1.4.3 and the extension group itself by the same letter, if this does not cause confustion. Note however that isomorphism of extensions stronger than simple isomorphism of the extension groups: in an isomorphism of extensions, the identiﬁcations of A as a subgroup of the extension groups correspond via the isomorphism of the extension groups, and similarly for G. The snake lemma shows that f is an isomorphism, so EXT(G, A) is a 0 0 groupoid. If (as,t) and (as,t) are 2-cocycles for E and E respectively, (f(as,t)) is another cocycle for E0, so the two extensions E and E0 yields the same element of H2(G, A). Conversely, given a factor system a we can construct an object of EXT(G, A) as follows. Let Ea = A × G and deﬁne a composition law on Ea by

s (x, s)(y, t) = (x yas,t, st) (3.1.4.7)

One then checks the following points:

• This composition law is a group law on Ea; the condition 5.2.2.6 is what is needed for the associativity law, and the identity is (a(1, 1)−1, 1).

• The map p : Ea → G defined by p(x, s) = s is surjective homomorphism with kernel isomorphic to A. Similarly the map i : A → Ea defined by i(x) = (xa1,1, 1) is injective and identifies A with the kernel of p.

• The cocycle assoiated to Ea by the section s 7→ (1, s) is a. 3.1. HOMOLOGY AND COHOMOLOGY 83

• The G-module structure of A deﬁned by the extension (Ea, i, p) by 3.1.4.4 coincides with the given G-module structure.

0 • If a ∼ a the Ea ' Ea0 , the isomorphism Ea → Ea0 being given by 0 (a, s) 7→ (abs, s) where a = adb. The verification of the first three points is easier if one uses the fourth to replace a by a normalized cocycle, i.e. one satisyfing a(1, 1) = 0. Details will be left as an exercise. It should be clear that the set of isomorphism classes of EXT(G, A) is a set, which we denote by Ext(G, A). We have shown that a 7→ Ea induces a bijection H2(G, A) −→∼ Ext(G, A). This shows that Ext(G, A) has a natural group structure; in fact it is induced by a “categorical group structure” on EXT(G, A) which we now describe. Let (E, i, p) and (F, j, q) be any two objects of EXT(G, A). The fibered product E × F is the set of (e, f) ∈ E×F such that p(e) = q(f); the composition G law (e, f)(e0, f 0) = (ee0, ff 0) makes it into a group. The identity 3.1.4.4 shows that the set of (i(a), j(a)−1) is a normal subgroup of E × F , and we denote by G E ∗ F the quotient of E × F by this subgroup. Denote by [e, f] the image of G (e, f) in E ∗ F , so that [i(a), f] = [e, j(a)f] for any a ∈ A. It is evident that • there is a surjective homomorphism r : E ∗ F → G such that r([e, f]) = p(e) = q(f), • the map k : A → E ∗ F defined by k(a) = [i(a), 1] = [1, j(a)] is injective and its image is the kernel of p. Thus (E ∗ F, k, r) is an object of EXT(G, A) which we denote by E ∗ F ; it is called the Baer sum of E and F . It is clear from the construction that if E ' E0 and F ' F 0 in EXT(G, A) then E ∗ F ' E0F 0, so the Baer sum induces a composition law on Ext(G, A). It is easily checked that the class in Ext(G, A) of the semidirect product AoG is an identity, and the class of (E, i, p) has as an inverse the class of (E, i−1, p). We have therefore defined a group law in Ext(G, A) and an isomorphism

Ext(G, A) −→∼ H2(G, A) (3.1.4.8) of abelian groups. It remains to check that this group law corresponds with the one on H2(G, A). Suppose that (E, i, p) and (F, j, q) be any objects of EXT(G, A) and let (E ∗ F, k, r) denote the Baer sum. If we choose sections (es), (fs) of p and q, the associated cocycles (as,t), (bs,t) are deﬁned by

eset = i(as,t)est, fsft = i(bs,t)fst.

Then the set of [es, fs] is a section of E ∗ F , and

[es, fs][et, ft] = [eset, fsft] = [i(as,t)est, j(bs,t)fst] = k(as,tbs,t)[est, fst] 84 CHAPTER 3. GROUP COHOMOLOGY

showing that (as,tbs,t) is indeed the cocycle associated to E ∗ F and the section ([es, fs]). The group H1(G, A) also has an interpretation relating to group extensions. If E is the extension 3.1.4.3 and α ∈ Aut(E), α(x)x−1 ∈ A since α induces the identity on G. Furthermore

α(xa)(xa)−1 = α(x)x−1 since α induces the identity on A. It follows that there is a map cα : G → A deﬁned by

−1 cα(s) = α(x)x for any x ∈ E such that p(x) = s.

If x, y ∈ E map to s and t respectively,

−1 cα(st) = α(xy)(xy) = α(x)α(y)y−1x−1 = (α(x)(α(y)y−1)α(x)−1)α(x)x−1 s = cα(s) · cα(t) by the deﬁnition of the G-action on A, so cα is an A-valued 1-cocycle. Finally if x maps to s then so does β(x) for any β ∈ Aut(E), and then

−1 −1 −1 cαβ(s) = (αβ)(x)x = α(β(x))β(x) β(x)x = cα(s)cβ(s).

1 It follows that α 7→ cα is a homomorphism Aut(E) → Z (G, A). We will leave it to the reader to check that this map is surjective, and the kernel is the 1 group InnA of inner automorphisms by elements of A. In particular H (G, A) is isomorphic to Aut(E)/InnA.

3.1.4.1 Pushouts. If f : A → B is a morphism of G-modules, the natural map H2(G, A) → H2(G, B) corresponds to the pushout extension. If E = (E, i, p) is an object of EXT(G, A), the pushout f∗E is the bottom row in the diagram

i p 1 / A / E / G / 1 (3.1.4.9)

f i2 1 / B / B qA E / G / 1 i1 p2 where B qA E is the amalgamted sum of the groups B and E over A, deﬁned as follows: since f is a G-module homomorphism,

N = {(f(a), i(a)) ∈ B × E | a ∈ A} ⊆ B × E is a normal subgroup of the product, and then

B qA E = (B × E)/N. (3.1.4.10) 3.1. HOMOLOGY AND COHOMOLOGY 85

If we denote the class of (b, e) ∈ B × E in B qA E by [b, e], the group law of B qA E is [b, e][c, f] = [bc, ef] and [f(a)b, e] = [a, i(b)e] for all a ∈ A. The maps i1, i2 and p2 in 3.1.4.9 are

i1(b) = [b, 1], i2(e) = [1, e] p2([b, e]) = p(e) respectively. If (es)s∈G is a section of p, ([1, es])s∈G is a section of p2. If (as,t) is the A-valued cocycle of E for the (es), the B-valued cocycle of f∗E for the ([1, es]) is (f(as,t)). Eﬀorts to similarly interpret Hn(G, A) for n > 2 quickly lead into the morass of higher category theory. So far as I know, only the case n = 3 has been worked out in the literature; it turns that H3(G, A) classiﬁes equivalence classes of certain grouplike monoidal categories.

3.1.5 Homology in degree zero and one. As we saw in sect 3.1.2 we can use the bar resolution to compute homology, but there seems to be no simple interpretation of these groups except in special cases. We mention two. For n = 0 we have

H0(G, A) ' Z ⊗Z[G] A (3.1.5.1) by deﬁnition. To interpret this, recall that the augmentation ideal I ⊂ Z[G] is the kernel of the augmentation map X X aug : Z[G] → Z ass 7→ as. (3.1.5.2) s∈G s∈G Then aug 0 → I → Z[G] −−→ Z → 0 (3.1.5.3) is exact by deﬁnition. It is easily checked that I is generated as a left or right ideal by the (s−1) for all s ∈ G. Tensoring the above short exact sequence with the G-module A yields X Z ⊗Z[G] A ' A/IA ' A/( (s − 1)A) = AG (3.1.5.4) s∈G This is a quotient of A as a G-module and a trivial G-module. But if B is any quotient of A on which G acts trivially, the quotient A → B must necessarily map (s − 1)a to 0 for any a ∈ A. It follows that A → B factors A → AG → B, or in other words AG is the largest quotient of A on which G acts trivially (just as AG is the largest submodule of A on which G acts trivially). The G-module AG is the module of coinvariants of A. To summarize, we have shown that

H0(G, A) ' AG (3.1.5.5) 86 CHAPTER 3. GROUP COHOMOLOGY for any G-module A. There seems to be no simple interpretation of H1(G, A) in general. When A G is a trivial G-module, i.e. A = A we have Hn(G, A) ' Hn(G, Z) ⊗Z A, and we will show that ab H1(G, Z) ' G (3.1.5.6) which implies that ab H1(G, A) ' G ⊗Z A (3.1.5.7) for any trivial G-module A. Observe ﬁrst that C1(G, Z) = Z1(G, Z) since d[s] ⊗ 1 = ([ ]s − [ ]) ⊗ 1 = 0. Next, the identity 0 = d[s|t] ⊗ 1 = ([s]t − [st] + [t]) ⊗ 1 = ([s] − [st] + [t]) ⊗ 1 shows that [s] ⊗ 1 + [t] ⊗ 1 = [st] ⊗ 1 (3.1.5.8) in H1(G, Z). We conclude that

α : G → H1(G, Z) s 7→ hsi ⊗ 1 mod boundaries ab is a surjective homomorphism. To show that H1(G, Z) ' G it suﬃces to show that G → H1(G, Z) has the universal property of the abelianization: any homomorphism f : G → A to an abelian group factors uniquely through α. Since B1 is a free Z[G]-module, B1 ⊗G Z is a free Z-module and there is a unique g : B1 ⊗G Z → A such that g([s] ⊗ 1) = f(s). ¯ Since f is a homomorphism 3.1.5.8 shows that g factors through f : H1(G, Z) → A and then f = f¯ ◦ α by construction. The factorization is unique since α is surjective.

3.1.6 Duality. Later we will need a generalization of 3.1.5.6. The homology groups Hn(G, Z) are computed by

Hn(G, Z) = Hn(P· ⊗G Z) for any projective resolution P· of Z. Therefore

Hom(Hn(G, Z), Q/Z) ' Hom(Hn(P· ⊗G Z, Q/Z)) n ' H (Hom(P· ⊗G Z, Q/Z)) n n ' H (Hom(P·, Q/Z) ' H (G, Q/Z) where the penultimate isomorphism holds because any G-module homomorphism Pi → Q/Z must factor through the G-coinvariants (Pi)G ' Pi ⊗G Z, the G-action on Q/Z being trivial. We have shown that n Hom(Hn(G, Z), Q/Z) ' H (G, Q/Z) (3.1.6.1) 3.1. HOMOLOGY AND COHOMOLOGY 87 and when n = 1 this says

1 Hom(H1(G, Z), Q/Z) ' H (G, Q/Z) ' Hom(G, Q/Z).

When G is a ﬁnite group this implies 3.1.5.6 by Pontryagin duality.

3.1.7 Cyclic groups. When G is a cyclic group there is a particular simple resolution of Z by free Z[G]-modules. Suppose G is generated by an element g of order d, let Cn be a free Z[G]-module of rank one and deﬁne : C0 → Z, dn : Cn+1 → Cn by

(x) = aug(x) ( (g − 1)x n odd (3.1.7.1) dn(x) = NG(x) n even. where d−1 d−2 NG(x) = (g + g + · + g + 1)x. (3.1.7.2)

It is easily checked that

d1 d0 → C2 −→ C1 −→ C0 −→ Z

∗ is exact. If we identify HomZ[G](Cn,A) ' A the maps dn : A → A are again given by the formula 3.1.7.1 (where now x ∈ A). Since Ker(g − 1) = AG and Im(g − 1) = IA, we ﬁnd that the Hn(G, A) are

H0(G, A) = AG ( (3.1.7.3) n Ker(NG)/IA n odd H (G, A) = G A /Im(NG) n > 0, n even

The same calculation can be made in homology, with the result

H0(G, A) = AG ( Ker(NG)/IA n > 0, n even (3.1.7.4) Hn(G, A) = G A /Im(NG) n odd.

3.1.8 Exercises.

3.1.8.1 Complete the arguments at the end of section 3.1.5 concerning the map Aut(E) → H1(G, A).

2 3.1.8.2 Using the exact sequence 3.1.5.3, show that H1(G, Z) ' IG/IG. 2 ab Then show directly that IG/IG ' G , thereby giving another proof of 3.1.5.6. 88 CHAPTER 3. GROUP COHOMOLOGY

3.1.8.3 Show that the isomorphisms 3.1.7.3, 3.1.7.4 are induced by the following chain map F : C· → B· where C· is the complex from section 3.1.7. If a is a generator of G and en is a Z[G]-basis of Cn then  [] n = 0  P i1 i2 is F (en) = I [a |a|a |a| · · · |a |a] i = 2s > 0,I = (i1, . . . , is) (3.1.8.1) P i1 i2 is I [a|a |a|a |a| · · · |a |a] i = 2s + 1,I = (i1, . . . , is)

3.1.8.4 Cyclic groups are not the only ﬁnite groups with periodic cohomology. Consider for example the generalized quaternion group Q4n which has generators x, y and relations

xn = y2, y4 = 1, xyx = y. (3.1.8.2)

Show that the trivial Q4n-module Z has a free resolution X· → Q4n with the following description:

• X4j free on one element aj, and

daj = Nej−1

P i where N = 0

0 • X4j+1 free on bj, bj with

0 dbj = (x − 1)aj, dbj = (y − 1)j;

0 • X4j+2 free on cj, cj with

0 0 0 dcj = Lbj − (y + 1)bj, dcj = (xy + 1)bj + (x − 1)bj

P i where L = 0

• X4j+3 free on ej with

0 dej = (x − 1)cj − (xy − 1)cj

Note that Q4n can be identiﬁed with the subgroup of the quaternion unit group H× by identifying x = eπi/n and y = j; in particular x and y have order 2n and 4 respectively.

n 3.1.8.5 Use the previous exercise to compute H (Q4n, Λ) for any trivial Q4n-module Λ. See [3, pages 253–4] if you need help. 3.2. CHANGE OF GROUP 89

3.2 Change of Group

n 3.2.1 Restriction and Inﬂation. The groups H (G, A), Hn(G, A) in addition to being functorial in A, also behave in predictable ways relative to G. The are for example functorial in G in the following sense: let f : G0 → G be a homomorphism, and for any G-module A let f ∗A be the G0-module deﬁned by “restriction to G0,” i.e. g0 ∈ G0 acts on A by a 7→ f(g0)a. With this notation there is a morphism

f ∗ : Hn(G, A) → Hn(G0, f ∗A) (3.2.1.1) natural in A. The easiest definition is by means of cocycles: if a(g1, . . . , gn) ∈ Zn(G, A) represents a class in Hn(G, A), the cochain f ∗(a) ∈ Zn(G0, f ∗A) defined by ∗ 0 0 0 0 f a(g1, . . . , gn) = a(f(g1), . . . , f(gn)) (3.2.1.2) is in Zn(G0, f ∗A), and simlarly if a ∈ Bn(G, A) then f ∗(a) ∈ Bn(G0, f ∗A). It follows that the f ∗ defined on Zn(G, A) passes to quotient by Bn(G, A), and this defines 3.2.1.1. In most of what follows we will write A in place of f ∗A when the meaning is obvious. Two cases of this are important. In the first H ⊆ G is a subgroup and the resulting map will be written

H n n ResG : H (G, A) → H (H,A). (3.2.1.3) and is called the restriction map, since if a ∈ Zn(G, A) represents a class in Hn(G, A), the restriction of a to Hn ⊆ Gn represents Res(a). The other is when H is a normal subgroup of G and π : G → G/H is the canonical projection. If A is a G-module, AH is naturally a G/H-module and the preceding construction yields a map

Hn(G/H, AH ) → Hn(G, AH ).

Composing this with the map Hn(G, AH ) → Hn(G, A) induced by the inclusion yields the inﬂation map

G n H n InfG/H : H (G/H, A ) → H (G, A). (3.2.1.4)

3.2.1.1 Proposition The sequence

0 → H1(G/H, AH ) −−→Inf H1(G, A) −−→Res H1(H,A) (3.2.1.5) is exact. Proof. (1) Inf is injective: pick a ∈ Z1(G/H, AH ) representing a class in H1(G/H, AH ) and suppose Inf(a) is a coboundary: a(π(g)) = gx − x for some x ∈ A and all g ∈ G. Since π(gh) = π(g) for any h ∈ H we must have ghx − x = gx − x, which implies hx = x for any h ∈ H. Thus x ∈ AH and it follows that a ∈ B1(G/H, AH ). 90 CHAPTER 3. GROUP COHOMOLOGY

(2) Res ◦ Inf = 0: again pick a ∈ Z1(G/H, AH ) representing a class in H1(G/H, AH ). Then a|H is constant: a(h) = a(1) for all h ∈ H. Since a|H is a 1-cocycle, a(h) = a(h1) = a(h) + ha(1) and it follows that a(h) = a(1) = 0. (3) Pick a ∈ Z1(G, A) representing a class in H1(G, A) and suppose a|H ∼ 0. This means that there is a c ∈ A such that a(h) = hc−c for all h ∈ H. Replacing a by a − d0(c), we may suppose that a(h) = 0 for h ∈ H. Then for g ∈ G and h ∈ H we have a(gh) = a(g) + ga(h) = a(g), i.e. a : G → A factors through a map G/H → A. Furthermore a(hg) = a(h) + ha(g) implies a(g) = ha(g), i.e a(g) ∈ AH for all g ∈ G. In other words a comes from an element of Z1(G/H, AH ).

3.2.1.2 Extensions again. Let’s interpret the restriction and inﬂation maps when n = 2 in terms of group extensions. For any homomorphism f : G → G0 the morphism 3.2.1.1 corresponds to the pullback of extensions. If E = (E, i, p) is any object of EXT(A, G), the pullback of E is the top row of the diagram

i f 0 p2 0 1 / A / E ×G G / G / 1 (3.2.1.6)

p1 f 1 / A / E / G / 1

0 where E ×G G is the ﬁbered product, p1 and p2 are the usual projections and ∗ if (a) = (i(a), 1). It is clear that f deﬁnes a functor f ∗EXT(G, A) → EXT(G0,A) (3.2.1.7) and we also use f ∗ to denote the corresponding map f ∗Ext(G, A) → Ext(G0,A). (3.2.1.8)

∗ 0 0 We denote the pullback by f E. Elements of E×GG are pairs (e, s) ∈ E×G such that p(e) = f(s), and the composition law is componentwise: (e, s)(f, t) = (ef, st).

0 If (es) is a section of p, (ef(s), s) is a section of E ×G G , and if the cocycle for ∗ E and the section (es) is (as,t), the correspondiing cocycle for f E is

∗ f (as,t) = (af(s),f(t)) (3.2.1.9) so that 3.2.1.8 corresponds to 3.2.1.1 under the identiﬁcation of Ext(G, A) with H2(G, A). In particular, the restriction map 3.2.1.3 corresponds to the functor

H ResG : EXT(G, A) → EXT(H,A) (3.2.1.10) arising from the pullback by the inclusion H,→ G. In this case the functor is simply H −1 ResG (E, i, p) = (p (H), i, p|H). (3.2.1.11) 3.2. CHANGE OF GROUP 91

Suppose now H is a normal subgroup of G. The construction of the inﬂation 3.2.1.4 shows that it arises from the functor

G H InfG/H : EXT(G/H, A ) → EXT(G, A) (3.2.1.12) which given an extension (E, i, p) of G/H by AH ﬁrst pulls it back by G → G/H and then pushes it out by AH → A (it is easy to check that performing these operations in the reverse order produces an isomorphic result). If we follow the recipe for the pushout and pullback we arrive at the description of the elements of the extension group: they can be identiﬁed with symbols

[a, e, s] with a ∈ A, e ∈ E, s ∈ G such that p(e) = π(s) (3.2.1.13) where π : G → G/H is the canonical homomorphism, and

[ab, e, s] = [a, i(b)e, s] for b ∈ AH . (3.2.1.14)

The group law is componentwise multiplication. Finally if (es)s∈G/H is a splitting of E and (as,t) is the corresponding cocycle, ([1, eπ(s), s])s∈G is a splitting G of InfG/H (E), and the cocycle for this splitting is (aπ(s),π(t)).

3.2.2 Coinﬂation and Corestriction. There is a similar result in homology. When H ⊆ G is a subgroup and M is a G-module we will call the change- G of-group map Hn(H,M) → HN (G, M) the corestriction, and denote it by CorH . The coinﬂation

G/H CoinfG : Hn(G, M) → Hn(G/H, MH ) (3.2.2.1) is deﬁned as the composition of the functoriality map Hn(G, M) → Hn(G, MH ) with the change-of-group map Hn(G, MH ) → Hn(G/H, MH ).

3.2.2.1 Proposition The sequence

Cor Coinf H1(H,M) −−→ H1(G, M) −−−→ H1(G/H, MH ) → 0 (3.2.2.2) is exact. The proof will be left as an exercise.

3.2.3 Shapiro’s Lemma. Suppose again that H ⊆ G is a subgroup and let A be an H-module. If M is an H-module we deﬁne G-modules

G G IndH (M) = Z[G] ⊗Z[H] M, CoindH (M) = HomZ[H](Z[G],M) (3.2.3.1)

G by means of the natural G-action on these G-modules. In the case of IndH (M) it is induced by the action of G on the left of Z[G], the right structure being G used to take the tensor product. In the case of CoindH (M) the left G-modules structure is used to form the Hom group, and the right action of G on Z[G] on itself induces a left action on the Hom group. 92 CHAPTER 3. GROUP COHOMOLOGY

3.2.3.1 Lemma The restriction of a projective left or right G-module to H is a projective H-module.

Proof. This is clear for the free G-module Z[G], a Z[H]-basis being given by a set of coset representatives of G/H. The result for free G-modules in general follows by taking direct sums, and for projective G-modules in general by passing to direct summands. We now recall the following bit of general algebra. Suppose R → A is a homomorphism of rings, M is a left A-module and N is a left R-module. We may use the left R-module structure of A to form the group HomR(A, N) and the right A-module structure of A to make HomR(A, N) a left A-module. The adjunction isomorphism is a canonical and functorial isomorphism

∼ HomR(M,N) −→ HomA(M, HomR(A, N)). (3.2.3.2) defined as follows. If f : M → N is R-linear and m ∈ M, the map fm : A → N defined by fm(a) = f(am) is R-linear and the map m 7→ fm is A-linear. This defines 3.2.3.2, and conversely if f : M → HomR(A, N) is given we set fm = f(m) for m ∈ M, and then m 7→ fm(1) is an R-linear map M → N. It is easily checked that this map is inverse to the previous one. Returning to the situation in which H is a subgroup of G, we take a projective G-resolution P· → Z, and note that P· is also a resolution of Z by projective H-modules, by the lemma. We now apply 3.2.3.2 with R, A, M and N replaced by Z[H], Z[G], P· and M respectively:

G HomH (P·,M) ' HomG(P·, HomZ[H](Z[G],M)) ' HomG(P·, CoindH (M)). From this we deduce the isomorphism called Shapiro’s lemma:

3.2.3.2 Theorem For any H-module M,

n n G H (H,M) ' H (G, CoindH (M)). (3.2.3.3) A G-module is M coinduced if

G M ' Coind1 (A) ' HomZ(Z[G],A) (3.2.3.4) for some abelian group A. Since the trivial group has no cohomology in positive degree, we ﬁnd

3.2.3.3 Corollary If M is a coinduced G-module, Hn(G, M) = 0 for all n > 0. Because of the corollary, coinduced modules play the same role for cohomology as free (or even ﬂat) modules do for homology. The following construct is used in the procedure called dimension-shifting, of which we have already seen an example. If M is a G-module and

∗ G M = Coind1 M = HomZ(Z[G],M) 3.2. CHANGE OF GROUP 93

∗ there is a natural map M → M which to m ∈ M assigns the map fm(g) = gm. This is a morphism of G-modules and we deﬁne M 0 by the exactness of

0 → M → M ∗ → M 0 → 0. (3.2.3.5)

By the corollary and the long exact sequence of cohomology we have

Hn(G, M 0) −→∼ Hn+1(G, M) (3.2.3.6) for all n > 0. This allows us to do inductive proofs on the degree.

3.2.3.4 Lemma Suppose H is a normal subgroup of G, M is a G-module and ∗ G ∗ ∗ H M = Coind1 (M). Then M is coinduced as an H-module and (M ) is coinduced as a G/H-module.

Proof. Since Z[G] is a free Z[H]-module there is an abelian group A and an isomorphism Z[G] ' Z[H] ⊗Z A of Z[H]-modules. Then ∗ M = HomZ(Z[G],M) ' HomZ(Z[H] ⊗Z A, M) ' HomZ(Z[H], HomZ(A, M)) and M ∗ is coinduced as an H-module. That (M ∗)H is a coinduced G/H-module follows from the isomorphism

H HomZ(Z[G],M) ' HomZ(Z[G/H],M).

The version of Shapiro’s lemma in homology is slighly simpler. With P· a projective G-resolution of Z as before, the extension of scalars isomorphism

P· ⊗Z[H] M ' P· ⊗Z[G] (Z[G]⊗Z[G]) shows that there is an isomorphism

n G Hn(H,M) ' H (G, IndH (M)) (3.2.3.7) for any H-module M. We could deﬁne an induced module to be one of the form G Ind1 (A), but these are the same as the free G-modules. We can now extend propositions 3.2.1.1 and 3.2.2.1.

3.2.3.5 Proposition Suppose H ⊆ G, n > 0 and M is a G-module such that Hi(H,M) = 0 for 0 < i < n. Then the inﬂation maps

Hi(G/H, M H ) −→∼ Hi(G, M) (3.2.3.8) are isomorphisms for 1 < i < n and the sequence

0 → Hn(G/H, M H ) → Hn(G, M) → Hn(H,M) (3.2.3.9) is exact. 94 CHAPTER 3. GROUP COHOMOLOGY

Proof. We argue by induction on n ≥ 0, where the case n = 1 is proposition ∗ G 3.2.1.1. Suppose the assertion is true for n − 1, let M = Coind1 (M) as before and deﬁne M 0 by 3.2.3.5. Since H1(H,M) = 0 the sequence

0 → M H → (M ∗)H → (M 0)H → H1(H,M) is exact. By lemma 3.2.3.4 M ∗ is coinduced both as a G-module and an H- module and M H is coinduced as a G/H-module. Then the isomorphisms 3.2.3.6 show that there is a commutative diagram

0 / Hi(G/H, (M 0)H ) / Hi(G, M 0) / Hi(H,M 0)

∂ ∂ ∂ 0 / Hi+1(G/H, M H ) / Hi+1(G, M) / Hi+1(H,M) in which the vertical maps are isomorphisms for i > 0. Since Hi+1(H,M) = 0 for i + 1 < n, Hi(H,M 0) = 0 for i < n − 1 and by induction the top row is exact. Then the bottom row is exact, which yields 3.2.3.11 and 3.2.3.12. The exact sequence 3.2.3.12 is the inﬂation-restriction sequence. The reader familiar with spectral sequences will realize that the proposition is an immediate consequence of a spectral sequence

p,q p q p+q E2 = H (G/H, H (H,M)) ⇒ H (G, M) (3.2.3.10) which is a special case of the spectral sequence of a composite functor, in this case the composition M G ' (M H )G/H . The corresponding result in homology is proven in the same way, we leave this as an exercise.

3.2.3.6 Proposition Suppose H ⊆ G, n > 0 and M is a G-module such that Hi(H,M) = 0 for 0 < i < n. Then the coinﬂation maps ∼ Hi(G, M) −→ Hi(G/H, MH ) (3.2.3.11) are isomorphisms for 1 < i < n and the sequence

Hn(H,M) → Hn(G, M) → Hn(G/H, MH ) → 0 (3.2.3.12) is exact.

3.2.4 The Corestriction. Suppose H ⊆ G is a subgroup and

· · · → Q2 → Q1 → Q0 → Z → 0 is a resolution of Z by projective left Z[H]-modules. Since Z[G] is a right ﬂat (in fact free) Z[H]-module, the sequence

· · · → Z[G] ⊗Z[H] Q2 → Z[G] ⊗Z[H] Q1 → Z[G] ⊗Z[H] Q0 → Z[G] ⊗Z[H] Z → 0 3.2. CHANGE OF GROUP 95

is exact, and is thus a resolution of Z[G] ⊗Z[H] Z by projective left G-modules. If M is a G-module, the adjunction isomorphism induces isomorphisms

∼ HomH (Q·,M) −→ HomG(Z[G] ⊗Z[H] Q·,M) from which we get a functorial isomorphism

n ∼ n H (H,M) −→ ExtG(Z[G] ⊗Z[H] Z,M) (3.2.4.1) for all n ≥ 0. The same argument applies if Q· → Z is a resolution by projective right Z[H]-modules, and then the isomorphisms

(Q· ⊗Z[H] Z[G]) ⊗Z[G] M ' Q· ⊗Z[H] M induce functorial isomorphisms

G Hn(H,M) ' Torn (Z ⊗Z[H] Z[G],M). (3.2.4.2) When H ⊆ G is a subgroup of ﬁnite index there is a morphism of left G- modules τ : Z → Z[G]⊗Z[H]Z deﬁned by choosing a set S of coset representatives of G/H and setting X τ(n) = x ⊗ n. (3.2.4.3) x∈S In fact for any g ∈ G, X X X gτ(n) = g x ⊗ n = σ(x)g ⊗ n = σ(x) ⊗ n = τ(gn) x∈S x∈S x∈S where σ is some permutation of S. Combining this morphism with the isomorphism 3.2.4.1 yields a functorial morphism

G n n CorH : H (H,M) → H (G, M) (3.2.4.4) called the corestriction. Similarly the map ρ : Z → Z ⊗Z[H] Z[G] deﬁned by X ρ(n) = n ⊗ x (3.2.4.5) x∈S is a morphism of right G-modules, which when combinened with 3.2.4.2 yields the corestriction H CorG : Hn(G, M) → Hn(H,M) (3.2.4.6) in homology. We see from the construction that that the corestriction is compatible with the connecting homomorphism:

0 → A → B → C → 0 96 CHAPTER 3. GROUP COHOMOLOGY is a short exact sequence of G-modules, the diagrams

n ∂ n+1 ∂ H (H,C) / H (H,A) Hn+1(H,C) / Hn(H,A) (3.2.4.7)

Cor Cor Cor Cor

n ∂ n+1 ∂ H (G, C) / H (G, A) Hn+1(G, C) / Hn(G, A) are commutative.

3.2.4.1 Proposition The for any subgroup H ⊆ G of ﬁnite index the composition Hn(G, M) −−→Res Hn(H,M) −−→Cor Hn(G, M) is multiplication by [G : H]. Proof. For n = 0, Cor ◦ Res is the composition

∼ HomG(Z,M) → HomH (Z,M) −→ HomG(Z[G] ⊗Z[H] Z,M) → HomG(Z,M). G An element of HomG(Z,M) is a function sending 1 7→ m for some m ∈ M . P Its composition with τ sends 1 7→ x∈S x ⊗ m, and the image of this under the adjunction sends 1 to X xm = |S|m = [G : H]m. x∈S The assertion is therefore true in cohomology for n = 0. The proof in homology is similar. If the assertion has been proven for n, apply 3.2.4.7 to the short exact sequence 3.2.3.5. Combining this with another obvious commutative square yields

∂ Hn(G, M 0) / Hn+1(G, M) .

Cor◦Res Cor◦Res

∂ Hn(G, M 0) / Hn+1(G, M)

The horizontal arrows are isomorphisms, and by induction the left vertical arrow is multiplication by [G : H]; the same is then true of the right one. Similarly for homology.

3.2.4.2 Corollary If G is a ﬁnite group then

|G|Hn(G, M) = 0 for all n > 0 and all G-modules M. Proof. Apply the proposition with H = 1. 3.2. CHANGE OF GROUP 97

3.2.4.3 Corollary If G is a ﬁnite group and M is a Q-vector space, Hn(G, M) = 0 for all n > 0.

Proof. By the last corollary it suﬃces to show that the Hn(G, M) are also Q- vector spaces. In fact this is true for the individual terms of HomG(B·,M), and so true for Hn(G, M) as well.

3.2.4.4 Example. If G is ﬁnite, the last corollary shows that the long exact sequence arising from 0 → Z → Q → Q/Z → 0 breaks up into isomorphisms

n−1 ∼ n ∂ : H (G, Q/Z) −→ H (G, Z) for all n > 1. When n = 2 the isomorphism H1(G, Q/Z) ' Hom(G, Q/Z) yields

∼ 2 ∂ : Hom(G, Q/Z) −→ H (G, Z) (3.2.4.8)

The formula for the coboundary shows that 3.2.4.8 is computed as follows: given a homomorphism f : G → Q/Z, choose for each s ∈ G a bs ∈ Z reducing to f(s) in Z/Z; then

∂f = (as,t) as,t = bs + bt − bst. (3.2.4.9)

The case G = Z/nZ will be particularly important later. The group Hom(G, Q/Z) has a canonical generator, which sends the class of i mod n to i/n ∈ Q/Z. For a splitting of Q → Q/Z we take for each a ∈ Q/Z the unique element of Q ∩ [0, 1) reducing to a. If on the other hand we identify elements of Z/nZ with their unique representatives in the interval [0, n), we ﬁnd ( i + j 0 i + j < n ai,j = = (3.2.4.10) n 1 i + j ≥ n.

We now have a means of constructing elements of H2(G, A) when A is any G-module: for any a ∈ AG there is a homomorphism f : Z → A of G-modules sending 1 to a. The induced homomorphism H2(G, Z) → H2(G, A) maps the class of the cocycle just constructed to ( 0 i + j < n f(ai,j) = (3.2.4.11) a i + j ≥ n.

The extensions of Z/nZ by A that arise from these cocycles have the following description; we will use multiplicative notation for A since this will look more natural for later applications. If in the notation of section 3.1.4 we set x = (1, 1), and identify A with a subgroup of the the extension E, then any element of E 98 CHAPTER 3. GROUP COHOMOLOGY has a unique expression uxi with u ∈ A and 0 ≤ i < n. The group law is then a consequence of the relations

xn = a, xb = sbx b ∈ A (3.2.4.12) where s : A → A is action of 1 ∈ Z/nZ on A.

3.2.4.5 Corollary If G is a finite group and M is a finitely generated G- module, Hn(G, M) is a finite group for all n > 0.

Proof. If M is ﬁnitely generated, the Hom(Bi,M) are ﬁnitely generated abelian groups for all n. The same is follows for the Hn(G, M), and the assertion follows from corollary 3.2.4.2.

We now explicate the corestriction in homology for n = 1 and M = Z, using ab the isomorphism H1(G, Z) ' G . Recall that this arises as the composition of ∂ the connecting homomorphism H1(G, Z) −→ H0(G, IG) and the isomorphisms 2 ab ab H0(G, IG) ' IG/IG ' G , where the latter identiﬁes the s in G with the 2 class of (s − 1) in IG/IG, or equivalently the class of 1 ⊗ (s − 1) ∈ Z ⊗ IG. Since Z[G] is free both as a right G-module and as a right H-module we may use the exact sequence 0 → IG → Z[G] → Z → 0 to compute both H1(G, Z) and H1(H, Z). This gives rise to a commutative diagram

H1(G, Z) / Z ⊗Z[G] IG

Cor Cor H1(H, Z) / Z ⊗Z[H] IG O

H1(H, Z) / Z ⊗Z[H] IH in which the top and bottom horizontal arrows are isomorphisms and the middle one is injective; this implies that Z ⊗H IH → Z ⊗H IG is injective. The right hand corestriction takes 1 ⊗ (s − 1) ∈ Z ⊗ IG to G X X (1 ⊗ x) ⊗ (s − 1) = 1 ⊗ (xs − x) H G H x∈H\G x∈H\G where the sum is over a set of right coset representatives. For every such x there is an hx(s) ∈ H such that xs = hx(s)σ(x) for some permutation of H \ G. Then

xs − x = hx(s)σ(x) − x

= hx(s)(σ(x) − 1) + (hx(s) − 1) − (x − 1) 3.3. TATE COHOMOLOGY 99

in IG, and

1 ⊗ (xs − x) = 1 ⊗ hx(s)(σ(x) − 1) + 1 ⊗ (hx(s) − 1) − 1 ⊗ (x − 1) H H H H

= 1 ⊗ (σ(x) − 1) + 1 ⊗ (hx(s) − 1) − 1 ⊗ (x − 1) H H H in Z ⊗ IG. Summing over x yields H X X 1 ⊗ (xs − x) = 1 ⊗ (hx(s) − 1) H H x∈H\G x∈H\G

Since Z ⊗H IH → Z ⊗H IG is injective we have proven the following formula for ab ab Cor : H1(G, Z) → H1(H, Q) viewed as a map G → H : choose a set of coset representatives x of H \ G and for s ∈ G ﬁnd a permutation σ of H \ G such that xs = hx(s)σ(x); then

Y ab Cor(s) = hx(s) in H (3.2.4.13) x where we have reverted to multiplicative notation. This is traditionally called the transfer in group theory, and sometimes written Ver : Gab → Hab (German Verlagerung); this usage is sometimes used in group cohomology.

3.3 Tate Cohomology

3.3.1 The Tate groups. When G is a finite group, Tate discovered that the cohomology and homology of a G-module M could be “spliced together” to obtain a set of functors Hˆ n(G, M) defined for all n ∈ Z, and with analogous formal properties (long exact sequence, restriction and corestriction...). The key is that the map X NG : M → MNG(m) = gm (3.3.1.1) g∈G G evidently takes its values in the submodule M . Its kernel contains IGM since the latter is generated by elements of the form (g − 1)m for all m ∈ M, which G are obviously annihilated by NG. Therefore NG defines a map MG → M and we define the groups Hˆ −1(G, M), Hˆ 0(G, M) by the exactness of

−1 NG G 0 0 → Hˆ (G, M) → MG −−→ M → Hˆ (G, M) → 0. (3.3.1.2)

−1 0 In particular Hˆ (G, M) is a subgroup of H0(G, M) and Hˆ (G, M) is a quotient of H0(G, M). The groups for n 6= −1, 0 are deﬁned by ( Hn(G, M) n > 0 Hˆ n(G, M) = (3.3.1.3) H−n−1(G, M) n < −1.

Observe that: 100 CHAPTER 3. GROUP COHOMOLOGY

0 G 0 (i). Hˆ (G, M) = 0 if and only if any m ∈ M can be written m = NG(m ) for some m0 ∈ M;

−1 (ii). Hˆ (G, M) = 0 if and only if any m ∈ M such that NG(m) = 0 is in IGM. If 0 → A → B → C → 0 is a short exact sequence the diagram

H1(G, C) / AG / BG / CG / 0

NA NB NC 0 / AG / BG / CG / H1(G, A) is commutative, and the serpent lemma yields an exact sequence

Hˆ −1(G, A) → Hˆ −1(G, B) → Hˆ −1(G, C) −→∂ Hˆ 0(G, A) → Hˆ 0(G, B) → Hˆ 0(G, C). (3.3.1.4) The map ∂ is computed in the usual way: given c ∈ Hˆ −1(G, C), lift it to G 0 a c¯ ∈ BG. Then NB(¯c) is an element of A and its image in Hˆ (G, A) is independent of the choice of c¯; this image is ∂(c). Combining this with the long exact sequences of homology and cohomology, we ﬁnd:

3.3.1.1 Theorem For any short exact sequence 0 → A → B → C → 0 there is a long exact sequence

→ Hˆ n(G, A) → Hˆ n(G, B) → Hˆ n(G, C) −→∂ Hˆ n+1(G, A) → (3.3.1.5) where for n 6= −1, ∂ is the coboundary for homology or cohomology. If G is a ﬁnite group there is an isomorphism

∼ ∗ C ⊗Z A −→ HomZ(C ,A) (3.3.1.6) for any finite free Z[G]-module C and abelian group A. The map 3.3.1.6 is defined by taking c ⊗ a to the function which to a homomorphism f : C → Z assigns the value f(c)a. For C = Z[G] this shows that induced and coinduced G-modules are one and the same. Furthermore if M is induced, Hˆ −1(G, M) = Hˆ 0(G, M) = 0; it suffices to check the case M = Z[G], which is easy. From this we see that the dimension- shifting technique used in regular homology and cohomology can be used in Tate cohomology as well. In fact any module is a quotient of an induced module and a submodule of a coinduced module, so the groups can be shifted either way. There is a version of the inflation-restriction exact sequence for the groups in degree 0 and −1: 3.3. TATE COHOMOLOGY 101

3.3.1.2 Lemma Suppose G is a ﬁnite group, H ⊆ G is a normal subgroup, and M is a G-module. If

Hˆ −1(H,M) = Hˆ 0(H,M) = Hˆ −1(G/H, M H ) = 0 then Hˆ −1(G, M) = 0.

Proof. We must show that if m ∈ M and NG(m) = 0 then m ∈ IGM. Now H ˆ −1 H NH (m) ∈ M and NG(m) = NG/H NH (m), so the vanishing of H (G/H, M ) implies that X NH (m) = (s − 1)ms s∈G/H

H where ms ∈ M , and the s are any set of representatives of G/H. Since Hˆ 0(H,M) = 0 we may then write

ms = NH (ns) with ns ∈ M. Then X NH (m) = (s − 1)NH (ns) s∈G/H X = NH ((s − 1)ns) s∈G/H where the second equality holds because H is normal in G. Therefore X NH (m − (s − 1)ns) = 0 s∈G/H

−1 and the vanishing of Hˆ (H,M) impliess that there are rt ∈ M for t ∈ H such that X X m − (s − 1)ns = (t − 1)rt. s∈G/H t Then X X m = (s − 1)ns + (t − 1)rt ∈ IGM s∈G/H t as required.

3.3.2 The double-bar complex Another definition of the Hˆ n(G, M) will be useful in establishing other properties of these groups. As we showed in ˇ section 3.1.2, Z has a resolution B· → Z by right G-modules whose terms are those of the bar resolution and with differentials given by 3.1.2.11. Now for any right G-module M, the Z-dual M ∗ = Hom(M, Z) has a left G-module structure arising from the right action of G on M, and if M → N is a homomorphism of 102 CHAPTER 3. GROUP COHOMOLOGY right G-modules, N ∗ → M ∗ is a homomorphism of left G-modules. Since G is ˇ finite and each Bn is free, the Z-linear dual

∗ d∗ d∗ ˇ∗ 0 ˇ∗ 1 ˇ∗ 0 → Z −→ B0 −→ B1 −→ B2 → (3.3.2.1) of the bar resolution is an acyclic complex of left G-modules. ˇ∗ For n < 0 we deﬁne Bn = B−n−1 and ( ∗ ◦ n = −1 dn = ∗ dn = d−n−1 n < −1.

Then the bar resolution and 3.3.2.1 can be spliced together at the Z term, yielding an inﬁnite acyclic complex

d1 d0 d−1 d−2 → B2 −→ B1 −→ B0 −−→ B−1 −−→ B−2 → (3.3.2.2) which we will call the double-bar complex. We will produce isomorphisms

ˆ n M M H (G, M) ' Ker(dn )/Im(dn−1) (3.3.2.3)

M for all n, where as before dn : HomG(Bn,M) → HomG(Bn+1,M). In fact for n > 0 the Bn and dn are the same as before, and 3.3.2.3. For n < −1 we ﬁrst observe that since the Hˆ 0 and Hˆ −1 of an induced module vanish, the homomorphism N G (C ⊗ M)G −→ (C ⊗ M) is an isomorphism for any ﬁnite free G-module C. From this we get an isomorphism

N G ∗ G ∗ C ⊗Z[G] M ' (C ⊗ M)G −→ (C ⊗ M) ' HomZ(C ,M) ' HomZ[G](C ,M) (3.3.2.4) and taking for C the terms of the bar resolution, further isomorphisms

ˇ∗ ˇ B−n−1 ⊗Z[G] M ' HomG(B−n−1,M) ' HomG(Bn,M) ∗ for all n < 0. For n < −1 these identify dn ⊗ idM = d−n−1 ⊗ idM , so again the isomorphism is obvious. The cases n = 0, −1 require a construction that we will need in greater generality later on. Recall that if M and N are free Z-modules, an isomorphism N → M ∗ corresponds to a nondegenerate pairing h , i : M × N → Z. If M (resp. N) is a right (resp. left) G-module the isomorphism N → M ∗ is G-linear if and only if hms, ni = hm, sni (3.3.2.5) for all s ∈ G. Suppose ﬁnally that M is free G-module with basis m1, . . . , mr, so that the mis for all i and s ∈ G are a Z-basis of M. There are n1, . . . , nr ∈ N ∗ −1 such that hmi, nji = δij, and the isomorphism N → M identiﬁes {s ni} with the dual basis to {mis}; in particular N is a free G-module. 3.3. TATE COHOMOLOGY 103

We will apply this construction to the modules Bn, Bˇn which are the free left and right G-modules on the set Gn. The pairing ˇ Bn × Bn → Z (3.3.2.6) deﬁned, for β, γ ∈ Gn by ( 1 β = γ, s = t−1 hβt, sγi = (3.3.2.7) 0 otherwise.

ˇ∗ ∼ satisfies the linearity condition 3.3.2.5, and the resulting isomorphism Bn −→ Bn will also be written ∗ ∼ Bn −→ B−n−1. (3.3.2.8) Note that in 3.3.2.7 we could replace the condition β = γ by β = π(γ) for any permutation π of Gn. The condition s = t−1 is of course essential. The dual of the augmentation : B0 → Z is X (1) = s[] s∈G since (1) must have inner product 1 with every []s. With the above identifications the differential d−1 : B0 → B−1 is X d[ ] = s[]. (3.3.2.9) s∈G

For any left G-module M the identiﬁcation HomG(Bˇ0,M) ' M associates to f : B0 → M the element f(1) ∈ M. If we set X ν = s ∈ Z[G] s∈G and use the previous identiﬁcation of B−1 with B0, the map HomG(B−1,M) → HomG(B0,M) induced by d−1 is the map induced by multiplication by ν, or in other words the norm map NG : M → M. Therefore the relevant part of the complex HomZ[G](B·,M), which is

HomZ[G](B−2,M) → HomZ[G](B−1,M) → HomZ[G](B0,M) → HomZ[G](B1,M) can be identiﬁed with

NG G B1 ⊗G M → MG −−→ M → HomG(B1,M) where the terms are in degrees −2 to 1. It follows that

−1 −1 0 0 H (HomG(B·,M) ' Hˆ (G, M),H (HomG(B·,M) ' Hˆ (G, M) as required. 104 CHAPTER 3. GROUP COHOMOLOGY

Later we will need the explicit formula for d−2 : B−1 → B−2, viewed as a map d : B1 → B0. If we write X d[ ] = as,ts[t] s,t∈G then

−1 −1 as,t = h[t]s , d[]i = hd([t]s ), []i = h[]ts−1, []i − h[]s−1, []i  1 s = t, s 6= 1  = −1 s = 1, t 6= s 0 otherwise.

Therefore d : B1 → B0 is X X d[ ] = (s − 1)[s] = (s − 1)[s]. (3.3.2.10) s6=1 s

3.3.3 Cyclic groups. We have seen that the homology and cohomology of a ﬁnite cyclic group is “nearly periodic” (equations 3.1.7.3, 3.1.7.4). The Tate groups in this case are completely periodic:

3.3.3.1 Proposition Suppose G is a ﬁnite cyclic group and A is a G-module. For all n ∈ Z, ( AG/N A n even Hˆ n(G, A) = G (3.3.3.1) Ker(NG)/IGA n odd. Proof. We have already proven this if n < −1 or n > 0, and for n = −1, 0 we use the exact sequence 3.3.1.2. Later it will be useful to have an explicit identiﬁcation

0 2 2 Hˆ (G, Z) ' Hˆ (G, Z) ' H (G, Z).

Recall from example 3.2.4.4 that if G ' Z/nZ, H2(G, Z) is cyclic of order n, generated by the class of the 2-cocycle ( 0 i + j < n a = (ai,j), ai,j = 1 i + j ≥ n where as before we identify i modulo n with an integer in the range [0, n). As in section 3.1.7, the isomorphisms 3.3.3.1 in positive degree are induced by the chain map F : C· → B· which in degree two is X F (e2) = [i|1] 0≤i

3.3.4 Products. Let’s recall some general results from homological algebra. For the moment let R be any ring and let A and B be (left) R-modules. An exact sequence 0 → B → En−1 → · · · → E0 → A → 0 (3.3.4.1) of length n+2 is called a Yoneda n-extension of A by B. These form a category in a manner similar to the category EXT(G, A) that was considered in section 3.1.4: morphisms are morphisms of complexes inducing the identity on A and B. Suppose now

0 → C → Fm−1 → · · · → F0 → B → 0 (3.3.4.2) is an m-extension of B by C. We can join this exact sequence to 3.3.4.1 at their common term B, leading to an (m + n)-extension

0 → C → Fm−1 → · · · → F0 → En−1 → · · · → E0 → A → 0 (3.3.4.3) which we will call the result of splicing together 3.3.4.2 and 3.3.4.1. If we denote the exact sequences 3.3.4.1 and 3.3.4.2 by E· and F· respectively we will denote by F· ◦E· the result of splicing the two extensions. This construction has obvious 0 functorial properties: a morphism u : E· → E· of n-extensions of A by B induces 0 0 a morphism F· ◦ E· → F· ◦ E· , and similarly a morphism u : F· → F· of m- 0 extensions of B by C induces a morphism F· ◦ E· → F· ◦ E·. Furthermore this operation is associative in an obvious sense: for any p-extension G· of C by D there is an isomorphism

(G· ◦ F·) ◦ E· ' G· ◦ (F· ◦ E·) in the category of (m + n + p)-extensions of A by D. n An n-extension 3.3.4.1 gives rise an element of ExtR(A, B) by the following well-known procedure: for any projective resolution P· → A of A there is a morphism of complexes

Pn+1 / Pn / Pn−1 / ··· / P0 / A / 0 (3.3.4.4)

f 0 / B / En−1 / ··· / E0 / A / 0 106 CHAPTER 3. GROUP COHOMOLOGY

that is unique up to homotopy. The map f : Pn → B deﬁnes an element of n ExtR(A, B) which is independent of the choice of resolution, and which we will call the class of the extension.1 m+n For any extensions E·, F· as above the class of F· ◦ E· in ExtR (A, C) m n depends only on the classes of F· and E· in ExtR (B,C) and ExtR(A, B). It follows that for any triple A, B, C of R-modules there is product operation

m n m+n ExtR (B,C) × ExtR(A, B) → ExtR (A, C) (3.3.4.5) which on the level of extensions is obtained by the splicing operation. Since splicing is associative up to isomorphism, the product 3.3.4.5 is associative as well. To show that 3.3.4.5 is well-deﬁned and explain how it is computed we choose projective resolutions P· → A, Q· → B,

· · · → P1 → P0 → A → 0

· · · → Q1 → Q0 → B → 0 of A and B, and denote them by P· and Q· (writing P−1 = A, Q−1 = B). The identity maps of A and B extend to morphisms of complexes P· → E·, Q· → F·, unique up to homotopy, and the classes of 3.3.4.1 and 3.3.4.2 are represented the homomorphisms f : Pn → A, g : Qm → B. Note that f factors through a ¯ 0 00 00 map f : Pn/dPn+1 → B and that P· = P· ◦ P· where P· is an n-extension of 0 A by Pn/dPn+1 and P· is a projective resolution of Pn/dPn+1. The situation is summarized by a commutative diagram which at the point of splicing is

% Pn+1 / Pn / Pn/dPn+1 / Pn−1 / Pn−2 f f¯ $ Q1 / Q0 / B / En / En−1 9

Since Pn is projective and Q0 → B is surjective we may ﬁll in the arrow Pn → Q0, and by a general result of homological arrow we may ﬁll in the remaining arrows to obtain a morphism of complexes P· → Q· ◦ E· which we will also denote by f·. 00 Since P· → E· is only unique up to homotopy, f· : P· → E· is likewise unique 0 0 up to homotopy; if f· is a another choice, corresponding to f : Pn → B we have 0 f = f + gd with d : Pn → Pn−1 and g : Pn−1 → B. Since Pn−1 is projective

1This is only one possible converntion; for example in Bourbaki [2] the class of the extension 3.3.4.1 would be (−1)n(n+1)/2f. This is done to improve the appearance of certain formulas. 3.3. TATE COHOMOLOGY 107

we may lift g to a map h : Pn−1 → Q0, and again by standard arguments this 0 extends to a homotopy between the maps f·, f· : P· → Q· ◦ E·. Thus f· : E· ◦ Q·, and composing this with Q· → F· yields a morphism of complexes P· → F· ◦E·. The term in degree m+n is a map Pn+m → C factoring through the quotient Pn+m/dPn+m+1, and this is the class of F· ◦ E·. As it is completely determined by the maps f : Pn → B, g : Qm → C, we have shown that 3.3.4.5 is well-deﬁned. In fact the argument shows that the product can be computed from any morphism of complexes

Pm+n+1 / Pm+n / ··· / Pn / Pn/dPn+1 / 0 (3.3.4.6)

g f¯ 0 / C / ··· / F0 / B / 0

¯ where as above f is the factorization of the map f : Pn → B giving the class of E·. In fact the argument produced one such morphism, and any other is homotopic to it.

3.3.4.1 Proposition Suppose that the class of the short exact sequence

0 → C → E → B → 0 (3.3.4.7)

1 is η ∈ ExtR(B,C). For any R-module C the map

n n+1 η∪ : ExtR(A, B) → ExtR (A, C) is the connecting homomorphism induced by 3.3.4.7. Proof. This follows from the above construction and the explicit description of the coboundary map. If we take R = [G] for any group G and recall Hn(G, A) = Extn ( ,A) Z Z[G] Z we get a product operation

Extm (A, B) × Hn(G, A) → Hm+n(G, B) (3.3.4.8) Z[G] which is associative in an obvious sense. To compute it we can use for P· in the above construction any projective resolution of Z, e.g. the bar complex. We will call this product the cup product (this is not standard terminology). We will write the operation 3.3.4.8 as (e, a) 7→ e ∪ a. Then proposition 3.3.4.1 yields:

3.3.4.2 Proposition Let G be a group and suppose

0 → B → E → A → 0 is a short exact sequence of G-modules. If e ∈ Ext1 (A, B) is the class of Z[G] this extension, then for any element a ∈ Hn(G, A), the cup product e ∪ a is the image of a under the coboundary ∂ : Hn(G, A) → Hn+1(G, B). 108 CHAPTER 3. GROUP COHOMOLOGY

For m ≤ n there is a similar product

m R R ExtR (B,C) × Torn (A, B) → Torn−m(A, C) leading to a “cap product” in group homology

m ExtR (B,C) × Hn(G, B) → Hn−m(G, C) but we will lead the details to the reader. This can be combined with the product 3.3.4.8 to get a cup product in Tate cohomology

Extm (A, C) × Hˆ n(G, A) → Hˆ m+n(G, C) (3.3.4.9) Z[G] for all m ≥ 0 and n ∈ Z. It is simpler however to proceed directly using the “double-bar” complex 3.3.2.2 and imitating the construction of 3.3.4.5 by means of 3.3.4.6. Suppose n ∈ Z and

0 → C → Fm−1 → · · · → F0 → A → 0 is an m-extension of A by C with class e ∈ Extm (A, C). An element α of Z[G] n Hˆ (G, A) is represented by a G-module map f : Bn → A vanishing on dBn+1. Since Bn is projective this lifts to a map Bn → F0 which extends to a morphism

Bm+n+1 / Bm+n / ··· / Bn / Bn/dBn+1 / 0

g f 0 / C / ··· / F0 / A / 0 of complexes. We deﬁne the cup product e ∪ α to be the class of the map m+n n n g : Bm+n → C in Hˆ (G, C). When n > 0, Hˆ (G, A) = H (G, A) and this construction is the same one as before. When m = 1 this is the coboundary in Tate cohomology; the proof is the same as that of proposition 3.3.4.1:

3.3.4.3 Proposition Let G be a group and suppose

0 → C → E → A → 0 is a short exact sequence of G-modules. If e ∈ Ext1 (A, C) is the class of this Z[G] extension, then for any element α ∈ Hˆ n(G, A), the cup product e ∪ α is the image of α under the coboundary ∂ : Hˆ n(G, A) → Hˆ n+1(G, C).

3.3.5 Duality in Tate cohomology. The duality formula 3.1.6.1 for homology and cohomology extends to a duality

−n−1 n Hom(Hˆ (G, Z), Q/Z) ' Hˆ (G, Q/Z) (3.3.5.1) for the Tate groups. In fact for n > 0 this is exactly the formula 3.1.6.1; the general case is proven in the same way, only replacing the resolution P· in section 3.1.6 by the double-bar complex B· and using 3.3.2.3. 3.4. GALOIS COHOMOLOGY 109

For positive n this duality can be expressed in terms of the cup product 3.3.4.9. In fact with A = Z and C = Q/Z, one case of 3.3.4.9 is

n −n−1 −1 Ext (Z, Q/Z) × Hˆ (G, Z) → Hˆ (G, Q/Z).

n Since n > 0, Ext (Z, Q/Z) = Hˆ n(G, Q/Z) by deﬁnition, while on the other hand 1 Hˆ −1(G, / ) = Ker(N )|( / ) = Ker(N )| / = / Q Z G Q Z G G Q Z |G|Z Z so the cup product is 1 Hˆ n(G, / ) × Hˆ −n−1(G, ) → / (3.3.5.2) Q Z Z |G|Z Z and we can replace the right hand side by Q/Z if we like since the Tate groups are killed by |G|.

[ﬁnish this]

3.4 Galois Cohomology

In this section L/K is a ﬁnite Galois extension and G is the Galois group Gal(L/K). Any abelian group M depending functorially on L is then a G- module, the most important cases being L and L×. In this section we focus on the cases when n = 0 or 1. The interpretation of H2(G, L×) will be discussed later.

3.4.1 General results. The computation of H0 is simple:

H0(G, L) = K,H0(G, L×) = K× and most importantly

ˆ 0 × × × H (G, L ) = K /NL/K L . (3.4.1.1)

For the additive group we have more generally, if perhaps disappointingly,

3.4.1.1 Proposition For n > 0,

Hn(G, L) = 0.

Proof. This follows from the normal basis theorem, which says says that L is an induced G-module, and thus a coinduced G-module since G is ﬁnite. The multiplicative case with n = 1 is known as Hilbert’s theorem 90 : 110 CHAPTER 3. GROUP COHOMOLOGY

3.4.1.2 Theorem With G and L as above,

H1(G, L×) = 0.

Proof. By Artin’s theorem on the linear independence of characters, the maps s : L× → L induced by the s ∈ G are linearly independent. Suppose now a(s) is a 1-cocycle with values in L×. By Artin’s theorem there is an x ∈ L× such that X b = a(s)s(x) 6= 0. s∈G Then for any s ∈ G, X X s(b) = (sa(t))st(x) = (a(st)a(s)−1)st(x) = a(s)−1b t∈G t∈G and thus a(s) = b · s(b)−1 is a coboundary. The version actually proven by Hilbert, however, was this:

3.4.1.3 Corollary Suppose L/K is a cyclic Galois extension and x ∈ L× is an element of norm 1. If s is a generator of Gal(L/K), x = s(y)y−1 for some y ∈ L×.

Proof. This follows from the theorem and the description of H1(G, M) in the cyclic case. Theorem 3.4.1.2 is the tip of a fairly large algebraic iceberg: Galois descent, Grothendieck’s theory of faithfully ﬂat descent and the Barr-Beck theorem in category theory are all descendents of this simple-looking result.

3.4.2 The case of algebraically closed residue field. The main result of this section is that if the residue field of K is algebraically closed, the G-module L× is cohomologically trivial. This is one of the key points in our treatment of local class field theory. Note that in this case the extension L/K is totally ramified.

3.4.2.1 Lemma Let K be a discretely valued nonarchimedean field with algebraically closed residue field. If L/K is a finite Galois extension with group G, L×, Hˆ 0(G, L×) = 0.

Proof. This follows immediately from theorem 2.4.2.5, since Hˆ 0(G, L×) = × × K /NL/K L .

3.4.2.2 Lemma Let K be a discretely valued nonarchimedean field with algebraically closed residue field. If L/K is a finite Galois extension with group G, L×, Hˆ −1(G, L×) = 0. 3.4. GALOIS COHOMOLOGY 111

Proof. The proof is by induction on the order of G, and the case of |G| = 1 is immediate. Since L/K is totally ramified G is solvable, and if it is nontrivial there is a normal subgroup H ⊂ G with G/H cyclic and nontrivial. If M is the fixed field of H, G/H ' Gal(M/K); then Hilbert’s theorem 90 and the periodicity of the cohomology of a cyclic group show that

Hˆ −1(G/H, (L×)H ) ' Hˆ −1(G/H, M ×) ' Hˆ 1(G/H, M ×) = 0.

On the other hand the induction hypothesis yields Hˆ −1(H,L×) = 0, and we know that Hˆ 0(H,L×) = 0 as well, so the lemma follows from lemma 3.3.1.2.

3.4.2.3 Corollary With the above hypotheses, the natural homomorphism

× × G (L )G → (L ) is an isomorphism.

Proof. This follows from lemmas 3.4.2.1, 3.4.2.2 and the exact sequence

−1 × × × G 0 × 0 → Hˆ (G, L ) → (L )G → (L ) → Hˆ (G, L ) → 0.

3.4.2.4 Theorem Let K be a discretely valued nonarchimedean field with algebraically closed residue field. If L/K is a finite Galois extension with group G, L× is cohomologically trivial, i.e. Hˆ n(G, L×) = 0 for all n.

Proof. The last two lemmas have shown this for n = 0 and −1, and the case n = 1 is theorem 90. If G is cyclic, the assertion follows from the periodicity of Tate cohomology of cyclic groups and the vanishing of Hˆ 0(G, L×) and Hˆ 1(G, L×). We ﬁrst treat the case n ≥ 2 by induction on |G|, the assertion being trivial if |G| = 1. When |G|= 6 1 we can ﬁnd a normal subgroup H ⊂ G with G/H cyclic and nontrivial. If Hi(G, L×) = 0 for all 1 < i < n the exact sequence of proposition 3.2.3.5 yields

0 → Hn(G/H, M ×) → Hn(G, L×) → Hn(H,L×) where M = LH is the ﬁxed ﬁeld of H. Then Hn(G/H, M ×) = 0 by the cyclic case and Hn(H,L×) = 0 by the induction hypothesis, and it follows that Hˆ n(G, L×) = Hn(G, L×) = 0 for all n ≥ 1. It remains to show that Hˆ n(G, L×) = 0 for n ≤ −2, or equivalently that × Hn(G, L ) = 0 for all n ≥ 1. Again we use induction on |G|. When n = 1 the exact sequence 3.2.2.2 is

× × × H1(H,L ) → H1(G, L ) → H1(G/H, (L )H ) → 0. 112 CHAPTER 3. GROUP COHOMOLOGY

By corollary 3.4.2.3 and the cyclic case,

× × H × H1(G/H, (L )H ) ' H1(G/H, (L ) ) ' H1(G/H, M ) = 0

H × × G where as before M = L . By induction H1(H,L ) = 0, so H1(G/H, (L ) ) = × 0 as well. Finally if Hi(G, L ) = 0 for 1 ≤ i < n, proposition 3.2.3.6 yields an exact sequence

× × × Hn(H,L ) → Hn(G, L ) → Hn(G/H, (L )H ) → 0.

× × H × × Again (L )H ' (L ) ' M and thus Hn(G/H, (L )H ) = 0 by the cyclic case, × × while Hn(H,L ) = 0 by induction. We conclude that Hn(G, L ) = 0.

3.4.2.5 Remark. The theorem also from a fairly general criterion for cohomological triviality due to Tate and Nakayama: a G-module M is cohomologically trivial if for every p-Sylow subgroup Gp ⊆ G there is an integer q such that q q+1 Hˆ (Gp,M) = H (Gp,M) = 0 (see [12, Ch. 9 §6 Th. 8] or [4, Ch. 4 §9 Th. 9]). This can be used in two ways. First, one can use lemma 3.4.2.1 0 × 1 × and Hilbert’s theorem 90, which shows that Hˆ (Gp,L ) = Hˆ (Gp,L ) = 0. Alternatively one could use the fact that Br(L/K) = 0 for any finite Galois extension of nonarchimedean fields whose residue fields are separably closed; then 1 × 2 × Hˆ (Gp,L ) = Hˆ (Gp,L ) = 0. Chapter 4

Central Simple Algebras and The Brauer Group

The determination of the Brauer group of local field is central to the cohomolig- ical approach to local class field theory (and correspondingly, the Brauer group of a global field in global class field theory). We will be more interested in the relative Brauer group attached to a finite Galois extension of local fields.

4.1 Central Simple Algebras

In this section we are concerned with some purely algebraic properties of central simple algebras. In this section k is a (commutative) ﬁeld, unless otherwise speciﬁed.

4.1.1 Basic Results. We first recall two basic results on division algebras, whose proofs can be found in the standard references. Recall that a K-algebra is a ring A and a ring homomorphism k → A such that the image of k is contained in the center of A. We say A is a central k- algebra if the image of k → A is the entire center of A. Finally a central simple k-algebra is a central k-algebra that is simple as a ring, i.e. its only 2-sided ideals are itself and the zero ideal. A central division algebra is automatically central simple. When k is a field, the structure map k → A of a k-algebra is injective. In this case A is naturally k-vector space and the ring operations are k-linear. The degree of the k-algebra A is the (possibly infinite) dimension of A as a k-vector space, and is denoted by [A : k] (just as for fields). The structure of a central simple k-algebra of finite degree is given by the following theorem. In the statement we use the convention that if A is a ring and M is a left A-module, then endomorphism ring EndA(M) acts on M on the right.

113 114 CHAPTER 4. THE BRAUER GROUP

4.1.1.1 Theorem (Wedderbun-Artin) A simple left artinian ring A is isomorphic to a matrix ring Mn(D), where D is the endomorphism ring of any simple left A-module.

4.1.1.2 Corollary Let k be a ﬁeld. A central simple k-algebra of ﬁnite degree over k is isomorphic to a matrix ring Mn(D) for some central division k-algebra D. Proof. Any such central simple algebra is artinian.

4.1.1.3 Corollary If k is an algebraically closed field, a central simple k-algebra of finite k-dimension is isomorphic to Mn(k). Proof. In fact k is the only division algebra of finite degree over k.

4.1.1.4 Theorem (Noether-Skolem) Suppose k is a ﬁeld, A is a central simple k-algebra of ﬁnite degree over k, B is a simple k-algebra and f, g : B → A are k-algebra homomorphisms. There is an a b ∈ A such that g = ad(b) ◦ f, i.e. such that g(x) = bf(x)b−1 for all x ∈ B. Since f and g are necessarily injective, this can be rephrased (in fact usually is) as follows: If B and B0 are any simple subalgebras of A, any isomorphism B → B0 extends to an inner automorphism of A.

4.1.1.5 Corollary If A is a central simple k-algebra of ﬁnite degree over k, any k-automorphism of A is inner. Proof. If α is an automorphism of A, apply the theorem with f = α and g = idA.

4.1.2 Products. From now on k is a field, unless otherwise specified. If A and B are k-algebras, the product algebra A⊗k B is the ordinary tensor product with its usual addition, and with multiplication defined by

(a ⊗ b)(a0 ⊗ b0) = aa ⊗ bb0. (4.1.2.1)

The map A → A ⊗k B deﬁned by a 7→ a is a k algebra homomorphism whose image we denote by A ⊗ 1. When k is a ﬁeld, A → A ⊗k B is an isomorphism of A onto A ⊗ 1. Likewise there is a k-algebra homomorphism B → A ⊗k B with image 1 ⊗ B. If A is a k-algebra and and S ⊆ A is any subset, we denote by CA(S) the centralizer of S. It is a sub-k-algebra of A.

4.1.2.1 Proposition Let A, A0 be k-algebras and B ⊆ A, B0 ⊆ A0 two k- subalgebras. Then

0 0 0 0 CA⊗kA (B ⊗k B ) = CA(B) ⊗k CA (B ). (4.1.2.2) 4.1. CENTRAL SIMPLE ALGEBRAS 115

0 0 0 0 Proof. It is clear that CA(B) ⊗k CA (B ) ⊆ CA⊗kA (B ⊗k B ), so it suﬃces to 0 0 0 0 show CA⊗kA (B ⊗k B ) ⊆ CA(B) ⊗k CA (B ). 0 0 We ﬁrst observe that CA⊗kA (B⊗1) ⊆ C(B)⊗k A . In fact if (ei)i∈I is a basis 0 0 of A as a k-vector space then any x ∈ A ⊗k A can be written uniquely as x = P i bi ⊗ei with bi ∈ B. Then for b ∈ B, x(b⊗1) = (b⊗1)x if and only if bbi = bib, 0 and if this holds for all b ∈ B then bi ∈ CA(B) and consequently x ∈ C(B)⊗k A . 0 0 0 This proves the stated inclusion, and by symmetry CA⊗kA (1⊗B ) ⊆ A⊗kC(B ). 0 0 0 0 0 Since CA⊗kA (B ⊗ B ) is a subring of CA⊗kA (B ⊗ 1) and CA⊗kA (1 ⊗ B ),

0 0 0 0 0 0 0 CA⊗kA (B ⊗ B ) ⊆ (C(B) ⊗k A ) ∩ (A ⊗ CA B ) = CA(B) ⊗k CA (B ).

The following corollaries are immediate:

4.1.2.2 Corollary The tensor product of two central k-algebras is central.

4.1.2.3 Corollary If D is a ﬁnite-dimensional division k-algebra, the diagonal embedding Z(D) → Z(Mn(D)) is an isomorphism.

Proof. Since Mn(D) ' Mn(k) ⊗k D, we have

Z(Mn(D)) = Z(Mn(k)) ⊗k Z(D) ' k ⊗k Z(D) ' Z(D)

∼ and the isomorphism k −→ Z(Mn(k)) is the diagonal embedding of k. Since the center of a division algebra is a ﬁeld, we get:

4.1.2.4 Corollary The center of a ﬁnite-dimensional simple k-algebra is a ﬁeld.

The product of two simple algebras is not necessarily simple, or even semisim- ple. We will see that this is the case, however if one is central simple. We will need the following bit of linear algebra. Let k be any division ring, V a left k-vector space and B = (ei)i∈I a basis of V (we do not assume V has ﬁnite P k-dimension). For any v ∈ V with v = i∈I aivi we denote by J(v) the set of i ∈ I such that ai 6= 0. Thus v = 0 if and only if J(v) = ∅. If W ⊆ V is a subspace, an element w ∈ W is primordial with respect to the basis B if at least one coeﬃcient of w relative to B is 1, and J(w) is minimal among the nonempty sets J(v) for all v ∈ W . For example if W = V , the primordial elements are the elements of B.

4.1.2.5 Lemma Let V , B and W ⊆ V be as above. (i) If w ∈ W is primordial and v ∈ W then J(v) = J(w) if and only if v is a multiple of w. (ii) W is spanned by its set of primordial elements.

Proof. (i) The condition is evidently suﬃcient, so suppose J(v) = J(w) and P w = i aiei with aj = 1. If bj is the coeﬃcient of ej in the expansion of v then 116 CHAPTER 4. THE BRAUER GROUP

J(v − bjw) is a proper subset J(w). Then v − bjw = 0 by the minimality of J(w). (ii) We will show that any v ∈ W is in the span of the primordial elements by induction on |J(v)|, and there is nothing to show if |J(v)| = 0. If |J(v)|= 6 0 pick w ∈ W with J(w) ⊆ J(v) with J(w) is minimal. We can assume that w is primordial; then for some c ∈ k, J(v − cw) is a proper subset of J(v). If v − cw = 0 we are done; otherwise we conclude by induction. We now revert to the situation in which k is a ﬁeld.

4.1.2.6 Lemma Suppose A is a k-algebra and D is a central division algebra over k. Any two-sided ideal a ⊂ A ⊗k D is generated as a left D-vector space by a ∩ A ⊗ 1. In particular, if A is simple then so is A ⊗k D.

Proof. The D-module structure of A ⊗k D is given by x(a ⊗ y) = a ⊗ xy, and we observe that a is a D-subspace of A ⊗k D. Choose a basis (ei)i∈I of A as a k-vector space; then B = (ei ⊗ 1)i∈I is a basis of A ⊗k D as a left D-vector P space. Suppose x ∈ a is primordial for a and B and write x = i ei ⊗ di with dj = 1. For any d ∈ D, X X x(1 ⊗ d) = ei ⊗ did = (1 ⊗ did)(ei ⊗ 1) i i and therefore J(x(1 ⊗ d)) = J(x). By (i) of lemma 4.1.2.5, x(1 ⊗ d) = (1 ⊗ d0)x 0 0 for some d ∈ D, and since dj = 1 we must have d = d. Then ddi = did, i.e. P P di ∈ Z(D) = k for all i, and x = i ei ⊗ di = i diei ⊗ 1 ∈ A ⊗ 1. For the last assertion it suﬃces to observe that the homomorphism A → A ⊗k D sending a 7→ a ⊗ 1 is nontrivial, and therefore injective, so we may identify a ∩ A ⊗ 1 with a two-sided ideal of A. This is either 0 or all of A, so a is either 0 or A ⊗k D.

4.1.2.7 Proposition If A and B are simple ﬁnite-dimensional k-algebras and at least one is central simple, A ⊗k B is simple.

Proof. We may suppose that B is central simple and write B = Mn(D) for some ﬁnite-dimensional k-division algebra D. Then

A ⊗k B = A ⊗k Mn(D) ' Mn(A ⊗k D) and by lemma 4.1.2.6, A ⊗k D is simple. Since it has ﬁnite dimension over 0 0 k, A ⊗k D ' Mm(D ) for some division k-algebra D and then A ⊗k B ' 0 Mmn(D ).

4.1.2.8 Theorem If A and B are central simple k-algebras of ﬁnite dimension then A ⊗k B is central simple. 4.1. CENTRAL SIMPLE ALGEBRAS 117

Proof. This follows from propositions 4.1.2.1 and 4.1.2.7. If A is a central simple algebra there is a natural k-algebra homomorphism

op A ⊗k A → Endk(A) (4.1.2.3)

op which maps a ⊗ b ∈ A ⊗k A to the k-linear map x 7→ axb.

4.1.2.9 Corollary The map 4.1.2.3 is an isomorphism of k-algebras.

op Proof. The theorem says that A ⊗k A is a simple k-algebra, and as 4.1.2.3 is not trivial it must be injective. Then

op 2 2 [A ⊗k A : k] = [A : k] = (dimk A) = dimk Endk(A) from which it follows that 4.1.2.3 must be surjective as well.

4.1.3 Base change. We now consider products where one factor is a ﬁeld.

4.1.3.1 Proposition Suppose A is a central simple k-algebra of ﬁnite dimension and K/k is any extension of ﬁelds. Then A ⊗k K is a central simple K-algebra.

Proof. We can write A = Mn(D) with D a central k-division algebra of ﬁnite dimension. Then A ⊗k K ' Mn(D ⊗k K), so it suﬃces to show that D ⊗k K is central simple over K. By lemma 4.1.2.6 D ⊗k K is a simple; by proposition 4.1.2.1,

Z(D ⊗k K) ' Z(D) ⊗k Z(K) ' k ⊗k K ' K and thus D ⊗k K is central simple. If A is a central simple k-algebra and K/k is an extension, we say that A splits over K if A ⊗k K ' Mn(K) for some n. For example, corollary 4.1.1.3 says that any central simple k-algebra of finite degree over k is split over the algebraic closure of k. We will eventually show that a central simple k-algebra of finite degree over k splits over a finite separable extension of k. If k is a local field, it splits over a finite unramified extension of k.

4.1.3.2 Corollary If A is a central simple k-algebra of ﬁnite degree, [A : k] is a square.

alg alg alg Proof. In fact [A : k] = [A ⊗k k : k ] and A ⊗k k is a matrix algebra over kalg by corollary 4.1.1.3.

4.1.3.3 Corollary A central simple k-algebra of ﬁnite degree splits over a ﬁnite extension of k. 118 CHAPTER 4. THE BRAUER GROUP

alg alg Proof. For any such algebra A, A⊗kk is split, and the isomorphism A⊗kk ' alg alg Mn(k ) only involves ﬁnitely many elements of k . If K is the ﬁnite extension they generate, A ⊗k K ' Mn(K).

op 4.1.3.4 Lemma Suppose B is a k algebra and let ` : B → Endk(B), r : B → Endk(B) be the embeddings given by left and right multiplication. Then `(B) and r(B) are each others’ centralizers in Endk(B). Proof. The centralizer of `(B) is the set of B-linear maps f : B → B where B is a left B-module in the usual way. If f(1) = b and x ∈ B then

f(x) = f(x · 1) = xf(1) = xb and consequently f ∈ r(Bop). Thus the centralizer of `(B) is contained in r(Bop), and the converse inclusion is clear. The centralizer of r(Bop) is treated similarly.

Note that the centralizer of `(B) in Endk(B) is EndB(B). In fact it is n op standard that Endk(B ) ' Mn(B ). The following is a version of what is sometimes called the double-centralizer theorem:

4.1.3.5 Theorem Let A be a central simple k-algebra of ﬁnite degree and B ⊆ A a simple sub-k-algebra. Then C = CA(B) is a simple k-algebra, B = CA(C) and [A : k] = [B : k][C : k]. (4.1.3.1)

Proof. The k-algebra D = A ⊗k Endk(B) is central simple over k by theorem 4.1.2.8. Consider the following two k-embeddings B → A ⊗k Endk(B): one is b 7→ b⊗1 by means of the given embedding of B in A, while the other is b 7→ 1⊗b where B → Endk(B) comes from the left action of B on itself. We denote the images of these embeddings by B1 and B2 respectively; then by Noether- Skolem the canonical isomorphism B1 ' B2 extends to an inner automorphism of D. From this it follows that CD(B1) ' CD(B2). Now by proposition 4.1.2.1, op C(B1) ' C ⊗k Endk(B), while C(B2) ' A ⊗k B by proposition 4.1.2.1 and lemma 4.1.3.4, so we conclude that

op C ⊗k Endk(B) ' A ⊗k B (4.1.3.2) and in particular

op [C ⊗k Endk(B): k] = [A ⊗k B : k]. This says [C : k][B : k]2 = [A : k][B : k] which yields 4.1.3.1. op op Since A is central simple and B is simple A⊗k B is simple by proposition 4.1.2.7, so by 4.1.3.2 C ⊗k Endk(B) is simple. A 2-sided ideal a ⊂ C produces a 4.1. CENTRAL SIMPLE ALGEBRAS 119

two-sided ideal a ⊗k Endk(B) in C ⊗k Endk(B), which is either 0 or the whole of C ⊗k Endk(B). Comparing dimensions shows that a = 0 or a = C, and we conclude that C is simple. We can now apply the above results with C in place of B, and if we set 0 0 0 C = CA(C) then [A : k] = [C : k][C : k]. It follows that [C : k] = [B : k], and 0 0 since evidently B ⊆ C , B = C = CA(C).

4.1.3.6 Corollary With the notation and assumptions of the proposition, if B is central simple over k then so is C, and the natural homorphism B ⊗k C → A is an isomorphism. Proof. Observe that B and C have the same center, which is B∩C, so C is central if B is. Then B ⊗k C is a simple algebra, and B ⊗k C → A must be injective. Since both sides have the same dimension, this map is an isomorphism. We can now characterize maximal commutative subalgebras of a central simple algebra:

4.1.3.7 Corollary Suppose A is a central simple k-algebra of ﬁnite degree and L ⊆ A is a subﬁeld containing k. The following are equivalent:

(i). L = CA(L), (ii). [A : k] = [L : k]2; (iii). L is a maximal commutative k-subalgebra of A. Proof. That (i) implies (ii) follows from theorem 4.1.3.5. (ii) implies (iii): let L0 ⊇ L be a commutative subalgebra containing L; then 0 L ⊆ CA(L) and consequently

2 0 2 [L : k] = [A : k] = [L : k][CA(L): k] ≥ [L : k][L : k] ≥ [L : k] .

Therefore [L : k] = [L0 : k] and L0 = L. (iii) implies (i): if x ∈ CA(L) is not in L then L[x] is a commutative k properly containing L, a contradiction.

4.1.3.8 Corollary If A is a central simple k-algebra of finite degree, any subfield of A that is a maximal commutative k-algebra splits A. In particular a subfield of A of degree p[A : k] splits A. Proof. Suppose L is a subfield of A that is a maximal commutative k-subalgebra. We can make A into an L-vector space via the natural action of L on the right of A, and for this vector space structure corollary 4.1.3.7 shows that [A : L] = [L : k]. On the other hand the action of A on itself by the left yields a k-algebra homomorphism A → EndL(A), whence an L-algebra homomorphism A ⊗k L → EndL(A) by adjunction. Since the latter homomorphism is nontrivial and A⊗k L is simple, A ⊗k L → EndL(A) is injective, and therefore an isomorphism since 120 CHAPTER 4. THE BRAUER GROUP both sides have the same L-dimension, namely [A : k] = [A : L]2. The last assertion follows from corollary 4.1.3.7.

Example. Suppose L/k is a finite extension of fields of degree n and let i : L → Endk(L) be the injective homomorphism given by left multiplication. Since 2 2 [Endk(L): k] = n = [L : k] , i embeds L as a maximal commutative subalgebra of Endk(L). We next show that any central simple k-algbra of finite degree over k splits over a finite separable extension.

4.1.3.9 Lemma Suppose k is a ﬁeld of characterstic p > 0 and D is a central k-division algebra. If every element of D is purely inseparable over K then D = K. Proof. Suppose a ∈ D \ K and choose a power q of p such that aq ∈ k. If σ = ad(a) then σ 6= 1 and σq = 1. Since K has characteristic p, (σ − 1)q = 0 and there is a positive integer f < q such that (σ − 1)f 6= 0. Choose f as large as possible and b ∈ D such that (σ − 1)f (b) 6= 0. We can now forget about q and set x = (σ − 1)f−1(b), y = (σ − 1)(x) = (σ − 1)f (b) then y 6= 0 and we put z = xy−1. Since (σ − 1)f+1(b) = 0, σ(y) = y and then

σ(z) = σ(x)σ(y)−1 = (x + y)y−1 = z + 1.

Once again there is a power q of p such that zq ∈ K, and then

zq = σ(zq) = σ(z)q = zq + 1 which is absurd.

4.1.3.10 Proposition Let D be a central division algebra over k of finite degree. There is a maximal commutative subalgebra of D that is a finite separable field extension of k. Proof. If k has characteristic zero, any finite extension of k is separable and the assertion follows from corollary 4.1.3.3. Suppose k has characteristic p > 0 and let F/k be a maximal separable extension of k in D. By the double centralizer theorem 4.1.3.5 the centralizer B = CD(F ) is simple, F = CD(B) and [D : k] = [B : k][F : k]. Then F ⊆ B and B is a central divison algebra over F . Since F is a maximal separable subfield of D every element of B is purely inseparable over F . Then B = F by lemma 4.3.1.2 and F is a maximal commutative subalgebra of D, i.e D splits over F . From this and the Artin-Wedderburn theorem we find:

4.1.3.11 Corollary Any central simple k-algebra splits over a ﬁnite separable extension of k. 4.2. THE BRAUER GROUP 121

4.2 The Brauer Group

We continue to assume k is a ﬁeld, and from now on all central simple k-algebras are assumed to have ﬁnite degree.

4.2.1 Brauer equivalence. If A and B are central simple k-algebras, we say that A and B are Brauer-equivalent or similar if A ⊗k Endk(V ) ' B ⊗k Endk(W ) for some ﬁnite-dimensional k-vector spaces V , W . We will write A ∼ B to mean that A and B are similar. This is a categorical equivalence relation: it is obviously reﬂexive and symmetric, while if A⊗k Endk(V ) ' B ⊗k Endk(W ) and B ⊗k Endk(X) ' C ⊗k Endk(Y ),

A ⊗k Endk(V ) ⊗k Endk(X) ' C ⊗k Endk(W ) ⊗k Endk(Y ).

Since this implies that

A ⊗k End(V ⊗k X) ' C ⊗k Endk(W ⊗k Y ) we see that A ∼ B and B ∼ C imply A ∼ C. It is easily checked that there is a set of equivalence classes of central simple k-algebras: for any n the number of maps kn × kn → kn is bounded by |k| if k is infinite, and is finite otherwise, so the number of isomorphism classes of k-algebras of degree n is bounded, and the same is true for the union of these sets for all n ≥ 1. We denote by [A] the class of A, and by Br(k) the set of equivalence classes of central simple k-algebras. A choice of basis of a k-vector space V of dimension n identifies Endk(V ) ' Mn(k), and then A ⊗k End(V ) ' Mn(A) for any k-algebra A. In fact the older definition of Brauer equivalence is that A ∼ B if and only if Mm(A) ' Mn(B) for some integers m, n.

4.2.1.1 Proposition (i) Two central division algebras over k are isomorphic if and only if they are similar. (ii) Two central simple k-algebras A, B are isomorphic if and only if A ∼ B and [A : k] = [B : k].

Proof. (i) Suppose D and E are central division algebras and Mn(D) ' Mm(E) for some m, n. Then any simple left Mn(D)-module is isomorphic to any simple left Mm(E)-module, and thus the endomorphism rings of these modules are isomorphic, i.e. D ' E. (ii) Suppose A ∼ B and write A ' Mn(D) and B ' Mm(E) with central divison algebras D, E. Then D ∼ A ∼ B ∼ E and consequently D ' E by (i). In particular [D : k] = [E : k], and since [A : k] = n2[D : k], [B : k] = m2[E : k] it follows that m = n. The converse is clear. From this point of view, theorem 4.1.1.1 and corollary 4.1.1.2 show that any central simple algebra is equivalent to a central division algebra which is unique up to isomorphism. Thus there is a bijection of Br(k) with the set of isomorphism classes of central division algebras over k. 122 CHAPTER 4. THE BRAUER GROUP

The real power of this construction comes from the fact that Br(k) has a natural group law. Suppose A ∼ A0 and B ∼ B0, say

0 0 0 0 A ⊗k End(V ) ' A ⊗k Endk(V ),B ⊗k End(W ) ' B ⊗k Endk(W ).

Then

0 0 0 0 (A ⊗k B) ⊗k Endk(V ⊗k W ) ' (A ⊗k B ) ⊗k Endk(V ⊗k W )

0 0 and consequently A ⊗k B ∼ A ⊗k B . Therefore ([A], [B]) 7→ [A ⊗k B] is a well-deﬁned binary operation on Br(k). It is associative by the associativity of tensor products, and [k] is an identity element. Finally, corollary 4.1.2.9 shows that [A][Aop] = [k], and we conclude that Br(k) is a group under this operation. It is called the Brauer group of k. Suppose K/k is a ﬁeld extension. We have seen that if A is a central simple k-algebra, A ⊗k K is a central simple K-algebra. Furthermore if A ∼ B then A ⊗k Endk(V ) ' B ⊗k (W ) and tensoring with K yields

(A ⊗k K) ⊗K EndK (V ⊗ K) ' (B ⊗k K) ⊗K EndK (B ⊗ K) so that A ⊗k K ∼ B ⊗k K. Finally, since

(A ⊗k K) ⊗K (B ⊗k K) ' (A ⊗k B) ⊗k K we see that A 7→ A ⊗k K deﬁnes a homomorphism Br(k) → Br(K). We deﬁne the relative Brauer group Br(K/k) to be the kernel of Br(k) → Br(K), so

0 → Br(k) → Br(K) → Br(K/k) (4.2.1.1) is exact by definition. Thus Br(K/k) is the group of equivalence classes of central simple algebras over k that are split by K. For the constructions of the next section we will need that fact that if K/k splits a central simple k-algebra A then A is similar to one in which K embeds as a maximal commutative subalgebra. Since a matrix algebra is isomorphic to its opposite algebra, a central simple k-algebra of finite dimension splits if and only if its opposite algebra does so. Thus the next proposition applies to any central simple algebra split by the finite extension K.

4.2.1.2 Proposition Suppose A is a central simple k-algebra, K/k is a ﬁnite op extension, V is a K-vector space and K ⊗k A ' EndK (V ) is an isomorphism. op Identify A and K with their images in Endk(V ) under the map EndK (V ) → op Endk(V ), and let B be the centralizer of A in Endk(V ). Then B is a central simple k-algebra similar to A, and K is a maximal commutative subalgebra of B. Proof. By corollary 4.1.3.6 B is central simple over k and the natural homo- op morphism B ⊗k A → Endk(V ) is an isomorphism. This shows that B ∼ A, and since K centralizes A in EndK (V ) ⊆ Endk(V ), K ⊆ B. 4.2. THE BRAUER GROUP 123

op The isomorphism A ⊗k K ' EndK (V ) shows that

2 [A : k] = [A ⊗k K : K] = dimK (V ) while B ⊗k A → Endk(V ) similarly shows that

2 2 2 [A : k][B : k] = dimk(V ) = dimK (V ) [K : k] .

Comparing these equalities shows that [B : k] = [K : k]2, and since K ⊆ B, K is maximal commutative in B.

4.2.2 Extensions and algebras. In the situations we will be concerned with K/k is a finite Galois extension with Galois group G, in which case the relative Brauer group can be computed by means of an isomorphism Br(K/k) ' H2(G, K×). However we recall from section 3.1.4 that the group of isomorphism classes of extensions of G by K× is also isomorphic to H2(G, K×). In this section we give a direct construction of an isomorphism Br(K/k) → Ext(Gal(K/k),K×). The natural way to construct an isomorphism Br(K/k) ' Ext(Gal(K/k),K×) would be to construct some kind of equivalence of categories and show that it is compatible with products. For this purpose we will rework the definition of Br(K/k) so as to make it the group of isomorphism classes of a catgory with a grouplike monoidal structure. Fix a finite extension K/k and consider the following category CS(K/k): objects are pairs (A, i) where A is a central simple k-algebra and i : K → A is a k-algebra homomorphism embedding K as a maximal commutative k- subalgebra of A. In particular [A : k] = [K : k]2 < ∞. A morphism f :(A, i) → (B, j) is a commutative diagram

i K / A . (4.2.2.1)

j B

Since A and B have the same dimension the k-algebra homomorphism A → B is an isomorphism. In other words CS(K/k), like EXT(G, K×) is a groupoid. The exact relation of CS(K/k) with Br(K/k) will be explained later. The functor E : CS(K/k) → EXT(G, K×) (4.2.2.2) is deﬁned as follows. For any object (A, i) in CS(K/k) let E(A, i) be the set of pairs (a, s) ∈ A× × G such that

i ◦ s = ad(a) ◦ i (4.2.2.3) or, more explicitly i(sx) = axa−1 (4.2.2.4) 124 CHAPTER 4. THE BRAUER GROUP for all x ∈ K and a ∈ A×. If (a, s) and (b, t) are elements of E(A, i) the computation

i ◦ st = i ◦ s ◦ t = ad(a) ◦ i ◦ t = ad(a) ◦ ad(b) ◦ i = ad(ab) ◦ i shows that (a, s)(b, t) = (ab, st) is a group law on E(A, i). The Noether-Skolem theorem shows that for any s ∈ G there is a a ∈ A× such that 4.2.2.3 holds, so (a, s) 7→ s deﬁnes a surjective homomorphism p : E(A) → G. The kernel of p is the subgroup of a ∈ A× centralizing the image of i, which is i(K×) since K is a maximal commutative subalgebra of A. Since (a, s) = (a, t) implies s = p(a) = t, the map (a, s) 7→ a is an injective homomorphism E(A) → A×. We will identify the map K× → A× induced by i with the map K× → E(A) sending x to (i(x), 1). Then p 1 → K× −→i E(A, i) −→ G → 1 is an object of EXT(G, K×) which we also denote by E(A, i) as before. There is no real ambiguity in using i for the k-algebra homomorphism K → A and the group homomorphism K× → E(A, i). In fact from 4.2.2.3 we see that

u : E(A, i) → A× (u, s) 7→ u (4.2.2.5) is an injective homomorphism, and the map i : K× → E(A, i) is the restriction to K× of the above u. We will call 4.2.2.5 the canonical embedding of E(A, i) in A×. Finally if f :(A, i) → (B, j) is a morphism in CS(K/k) we deﬁne E(f) to be the evident morphism of extensions

i 1 / K× / E(A, i) / G / 1

f 1 / K× / E(B, j) / G / 1 j induced by f : A → B. This completes the construction of 4.2.2.2.

4.2.3 Crossed products. A construction called the crossed product algebra will yield an inverse functor to 4.2.2.2. We consider the following data: G is a group, R is a commutative ring with a G-module structure and E is an object

p 1 → R× −→i E −→ G → 1 (4.2.3.1) of EXT(G, R×). Given this,

R× × (R × E) → R × E (a, (x, u)) 7→ (xa−1, i(a)u) 4.2. THE BRAUER GROUP 125 deﬁnes an action of R× on R×E, and we denote by [R,E] the quotient of R×E by this action. If (x, u) ∈ R × E we denote by [x, a] its image in [R,E]. Since ai(x) = i(p(a)x)a in E, the set [R,E] has a monoid structure for which

[x, a][y, b] = [x · p(a)y, ab]. (4.2.3.2)

For s ∈ G let [R,E]s be the set of [x, a] ∈ [R,E] such that p(a) = s. The monoid structure clearly induces maps

[R,E]s × [R,E]t → [R,E]st. (4.2.3.3)

If e0 ∈ E is such that p(e0) = s, any element of [R,E]s has a unique expression as [x, e0]. In particular if s = 1, x 7→ [x, 1] is a canonical isomorphism ∼ R −→ [R,E]1 of monoids for the operation of multiplication on R. If we use this identiﬁcation to make [R,E]1 into a commutative ring, the addition operation is [x, 1] + [y, 1] = [x + y, 1].

More generally if s ∈ G and p(e) = s, any element of [R,E]s has a unique expression as [x, e] and we can then identify [R,E]s with a free R-module of rank one, the operations being

[x, e] + [y, e] = [x + y, e] a[x, e] = [a, 1][x, e] = [ax, e].

It is easily checked that the R-module structure on [R,E]s is independent of the choice of e. Finally, the free R-module M R E = [R,E]s (4.2.3.4) s∈G has a natural ring structure arising from 4.2.3.3; the additive and multiplicative identities are [0, 1] and [1, 1] respectively, and any element of the form [x, e] with × x ∈ R is a unit. We call R E the crossed product algebra associated to the extension E (note that we are using the symbol E to denote both a group and an extension; the construction R E depends on the extension and not just on the group). Note how R E diﬀers from the set [R,E]: in R E any element of the form [0, e] is the additive identity of R E, but two elements [0, e], [0, t] ∈ [R,E] with p(e) 6= p(f) are distinct in [R,E]. In geometric terms, if the individual summands in 5.2.2.5 are thought of as “coordinate axes” then [R,E] is the disjoint sum of the coordinate axes. The equality 4.2.3.2 shows that

ιE : R → R E ιE(a) = [a, 1]1 (4.2.3.5) is a homomorphism of R onto a subring of R E, the canonical embedding of R. If k = RG ⊆ R is the subring of G-invariants of R, we see from 4.2.3.2 that 126 CHAPTER 4. THE BRAUER GROUP i maps k into the center of R E, so that R E is an k-algebra. Likewise the group homomorphism

× υE : E → (R E) υE(a) = [1, a]p(a) (4.2.3.6) will be called the canonical embedding of E. By construction the diagram

× × K ιE |K (4.2.3.7) i + K E 3 E υE is commutative. To do computations in a crossed product algebra it helps to observe that by construction any a ∈ R E has a unique expression X a = as with as ∈ [R,E]s. (4.2.3.8) s

If we identify x ∈ R with its image in R E by the canonical embedding the formula 4.2.3.2 shows that s asx = xas (4.2.3.9) for all x ∈ R. An isomorphism (E, i, p) −→∼ (E0, i0, p0) in EXT(G, R×) induces an isomor- ∼ 0 phism R E −→ R E of crossed product algebras. Thus E 7→ R E deﬁnes a functor × EXT(G, R ) → Algk. (4.2.3.10) The crossed product has the following universal property:

4.2.3.1 Proposition Suppose (E, i, p) is an extension of G by R×, k = RG, B is a k-algebra, j : R → B is a k-algebra homomrphism and u : E → B× is a group homomorphism such that

j(x) = u(i(x)) and u(a)j(x)u(a)−1 = j(p(a)x) (4.2.3.11)

× for all x ∈ R and a ∈ E. There is a unique homorphism f : R E → B such that j|K×

× × R ιE |K (4.2.3.12) f # i + R E B 3 / ; E υE

u is commutative. 4.2. THE BRAUER GROUP 127

Proof. Uniqueness holds because R E is generated as a k-algebra by elements of the form ι(x) and υ(a). For existence, observe that the conditions of 4.2.3.1 imply that there is an R-linear map [R,E]s → B such that

fs([x, a]s) = j(x)u(a) and the diagram

[R,E]s × [R,E]t / [R,E]st

fs×ft fst B × B / B is commutative. The map M M f = fs : [R,E]s → B s s is then a k-algebra homomorphism R E → B satisfying the given conditions. Suppose that (E, i, p) is an object of EXT(G, R×) and f : G0 → G is a homomorphism. As before denote by f ∗E = (E0, i0, p0) the pullback extension

i0 p 1 / R× / E0 / G / 1

g 1 / R× / E / G / 1 i p and recall that the G0-module structure of A is the one induced by f. If u : × 0 E → (R E) is the canonical embedding then u ◦ g is a homomorphism E → × (R E) satisfying the conditions of proposition 4.2.3.1, so there is a unique R-algebra homomorphism

∗ 0 f : R E → R E (4.2.3.13)

0 ∗ 0 0 0 0 such that u◦(g|E ) = f ◦u where u : E → R E is the canonical embedding of the pullback. Suppose on the other hand that R0 is a ring with a G-action and f : R → R0 is a ring homomorphism compatible with the G-actions. If (E, i, p) is an object of EXT(G, R×) pushout extension by f is

i p 1 / R× / E / G / 1

f g 1 / (R0)× / E0 / G / 1 p0 128 CHAPTER 4. THE BRAUER GROUP

0 0 0 and we may form R E . There is a canonical R -algebra isomorphism 0 ∼ 0 0 f∗ : R ⊗R (R E) −→ R E (4.2.3.14)

× 0 0 × identifying the canonical embeddings u : E → (R E) and u : E → (R E) . As before this follows from the universal property of crossed products.

4.2.4 Application to the Brauer group. We now return to our original situation.

4.2.4.1 Theorem Suppose K/k is a ﬁnite Galois extension and E is an ex- × tension of G = Gal(K/k) by K . The k-algebra K E is central simple, and ι : K → K E embeds K as a maximal commutative subalgebra. P Proof. K E is central: suppose x = s as is in the center, where as before as ∈ [K,E]s, and let ` ∈ K be a primitive element. Then

X X s `as = `x = x` = `as s s

s and since ` 6= ` for s 6= 1 we must have as = 0 for s 6= 1. Therefore x = a1u1 ∈ s G K. Then xus = usx = xus for all s ∈ G implies x ∈ K = k. K E is simple: pick a basis {us}s∈G of K E with us ∈ [K,E]s and let I ⊆ K E be a nonzero ideal. It suﬃces to show that in the K-vector space K E with basis {us}s∈G, a primordial element of I is one of the us, for if us ∈ I then so is us−1 us, which is a nonzero multiple of the identity. P Suppose that x = s xsus is primordial and ss, st 6= 0. If b ∈ K is a primitive element then x0 = s(b)x − xb is a nonzero element of I since s(b) 6= t(b). Then 0 |J(x )| < |J(x)| contradicts the minimality of J(x). Therefore x = us for some s ∈ G. 2 Since |G| = [K : k], [K E : k] = [K : k] and i(K) is a maximal commutative subalgebra of K E. The theorem shows that E 7→ (K E, ι) deﬁnes a functor CS : EXT(G, K×) → CS(K/k). (4.2.4.1)

4.2.4.2 Theorem The functors

CS : EXT(G, K×) → CS(K/k) E : CS(K/k) → EXT(G, K×) are inverse equivalences of categories. Proof. (1) We show CS ◦ E is isomorphic to the identity functor of CS(K/k), or in other words there is a functorial isomorphism (K E(A, i), ι) ' (A, i) for every (A, i) in CS(K/k). If u : E(A, i) → A× is the canonical embedding, i = j and u satisfy the conditions of proposition 4.2.3.1, so there is a k-algebra homomorphism f : K E(A) → A which is necessarily injective since K E(A) is simple. Since the dimensions of the source and target are the 4.2. THE BRAUER GROUP 129 same, f is an isomorphism. It is immediate that f extends to an isomorphism (K E(A, i), ι) ' (A, i) which is functorial in A. (2) We show E ◦ CS is isomorphic to the identity functor of EXT(G, K×). × × Let (E, i, p) be an object of EXT(G, K ). For s ∈ G the set [K,E]s = [K,E]s \{[0, 1]} is exactly the set of a ∈ K E satisfying 4.2.2.3 for i = ι. × × If we denote by [K,E] the disjoint union of the [K,E]s endowed with the × multiplication induced by that of K E, EXT (CS(E)) is group [K,E] en- × × dowed with the injection i : K → [K,E] induced by ι : K → K E and the projection [K,E]× → G given by [x, e] 7→ p(e). On the other hand the group homomorphism E → [K,E]× given by e 7→ [1, e] is clearly an isomorphism, and extends to a isomorphism of extensions E → EXT (CS(E)). It is clear from the construction that this is functorial in E.

4.2.4.3 Corollary For any ﬁnite Galois extension L/K with Galois group G there are canonical isomorphisms

Br(L/K) −→∼ EXT(G, L×) −→∼ H2(G, L×) (4.2.4.2) of groups, where the ﬁrst is induced by the functor E and the second is 3.1.4.8 for A = L×.

Since EXT(G, K×) has a “product structure” (more formally, a monoidal structure) arising from the Baer sum of extensions we expect that CS(K/k) should have a similar structure that is in some way compatible with the product in Br(K/k). The diﬃculty is that if (A, i) and (B, j) are in CS(K/k), the tensor product A ⊗k B does not have an embedding of K as a maximal commutative subalgebra, since the dimensions are wrong except in the trivial case where A or B is k. We must therefore use the procedure of proposition 4.2.1.2. Suppose (A, i) and (B, j) are objects of CS(K/k) and denote by VA and VB be the K-vector spaces A and B with the K-vector space structure coming from left multiplication by elements of K via i and j respectively. By comparing dimensions we see that there is an isomorphism

op op ∼ K ⊗k A ⊗k B −→ EndK (VA ⊗K VB) (4.2.4.3) sending c ⊗ a ⊗ b to the K-linear endomorophism

x ⊗ y 7→ i(c)xa ⊗ yb = xa ⊗ j(c)yb.

op op Denote by A ∗k B the centralizer of the image of A ⊗k B in Endk(VA ⊗k VB) and by i ∗ j the embedding K → A ∗k B induced by 4.2.4.3 and the inclusion of EndK (VA ⊗K VB) into Endk(VA ⊗K VB). By proposition 4.2.1.2, A ∗k B is a central simple k-algebra similar to A ⊗k B and i ∗ j embeds K → A ∗k B as a maximal commutative subalgebra; thus (A ∗k B, i ∗ j) is an object of CS(K/k). If (A, i) ' (A0, i0), (B, j) ' (B0, j0) 130 CHAPTER 4. THE BRAUER GROUP are isomorphisms in CS(K/k) it is clear from the construction that there is

0 0 0 0 (A ∗k B, i ∗ j) ' (A ∗k B , i ∗ j ) as well.

4.2.4.4 Proposition For any two objects E, F of EXT(G, K×),

∼ CS(E) ∗k CS(F ) −→ CS(E ∗ F ). (4.2.4.4) in CS(K/k).

Proof. Recall the construction of E ∗ F : as a group it is the cokernel of the ﬁbered product E × F by the image of the subgroup of elements of the form G (i(x), j(x)−1). We denote by [e, f] the image of (e, f) ∈ E × F in E ∗ F . The G projection E ∗ F → G is (e, f) 7→ p(e) = q(f) and the kernel w : K× → E ∗ F is

w(x) = [i(x), 1] = [1, j(x)].

× × Set A = K E, B = K F and let u : E → A , v : F → B be the canonical injections, and denote by i : K → A, j :→ B the canonical injections. Finally let VA, VB be as in the discussion preceding the proposition. For any (e, f) ∈ E × F , G

x ⊗ y 7→ u(e)x ⊗ v(f)y defines a k-linear endomorphism of VA ⊗K VB (that this definition is consistent with the identities i(a)x ⊗ y = x ⊗ j(a)y follows from p(e) = q(f)). Since it is op op defined by left multiplication, this endomorphism centralizes A ⊗k B and × thus belongs to A ∗k B, whence a homomorphism E × F → (A ∗k B) . It is G −1 clear that elements of the form (i(a), j(a ) act trivially on VA ⊗k VB, so this × homomorphism passes to the quotient, yielding w : E ∗ F → (A ∗k B) . Now the embedding ` : K× → E ∗ F is a 7→ (i(a), 1) = (1, j(a)), and it is clear from the construction that (e, f)`(a) = `(sa)(e, f) if p(e) = q(f) = s. It follows from the universal property of crossed product algebras that w and ` define a (necessarily injective) k-algebra homomorphism K (E ∗ F ) → A ∗k B, and since source and target have the same degree |G| over k, it is an isomorphism ∼ K (E ∗ F ) −→ A ∗k B as required. We can now explain the precise relation between CS(K/k) and the Brauer group. Denote by π0(CS(K/k)) the set of isomorphism classes of objects of CS(K/k). If (A, i) is an object of CS(K/k) we denote by [A, i] the corresponding element of π0(CS(K/k)). The above results show that the product ∗k induces a binary composition law on π0(CS(K/k)). 4.2. THE BRAUER GROUP 131

4.2.4.5 Proposition The set π0(CS(K/k)) with the binary composition law induced by ∗k is a group, and there is a unique homomorphism of groups α : π0(CS(K/k)) → Br(k) such that

α([A, i]) = [A] (4.2.4.5) for all (A, i) in CS(K/k). The homomorphism α is injective and its image is Br(K/k). Proof. Uniqueness of the homomorphism is clear, and that α is compatible with products follows from the fact that A ∗k B ∼ A ⊗k B. It is well-deﬁned since (A, i) ' (B, j) implies that A ' B and thus A ∼ B; furthermore if A is any central simple k-algebra A split by K, [A] is in the image of α since it is similar to one with an embedding of K as a maximal commutative subalgebra, and the converse is clear. Thus the image of α is Br(K/k). If α(A, i) = α(B, j) then A ∼ B and hence A ' B since A and B have the same dimension (given that K is embedded as a maximal commutative subalgebra of both). By the Noether-Skolem theorem there is an inner automorphism u of B such that j = u ◦ i, and this u is an isomorphism (A, i) ' (B, j) in CS(K/k). It follows that α is injective, and since its image is a subgroup of Br(K/k), π0(CS(K/k)) is a group. The procedure for associating a 2-cocycle to an element of Ext(G, K×) can thus be applied to Br(K/k). Recall that for any object (E, i, p) of EXT(G, K×) we choose es ∈ E for all s ∈ G such that p(es) = s. For all s, t ∈ G, p(eset) = × st = p(est) and thus there is an as,t ∈ K such that

eset = as,test.

× ∼ Then (as,t) is a 2-cocycle and E] 7→ (ast) induces an isomorphism Ext(G, K ) −→ H2(G, K×). In the crossed product algebra K o E we may then set us = [1, es], and then

s usut = as,tust, usi(a) = i( a)us (4.2.4.6) for all a ∈ K. Conversely, if A is any central simple k-algebra and i : K → A is a k-embedding of K as a maximal commutative subalgebra, there are us ∈ A × satisfying the second equality of 4.2.4.6, and then as,t ∈ K satisfying the ﬁrst. 2 × The class of the 2-cocycle (as,t) in H (G, K ) corresponds to the class of A in Br(K/k) by the previous isomorphism. In fact the following slightly more general statement is true:

4.2.4.6 Proposition Suppose A is a k-algebra, K/k is a ﬁnite Galois extension with group G and i : K → A is a k-algebra homomorphism. Suppose that for all s ∈ G there are elements us ∈ A satisyﬁng 4.2.4.6 and which generate A when it is made into a K-vector space via i. Then A is a central simple k-algebra 2 × split by K and the class (as,t) in H (G, K ) corresponds to the class of A in Br(K/k). 132 CHAPTER 4. THE BRAUER GROUP

Proof. Let E be an extension whose class is that of the cocycle (as,t). The universal property of crossed products shows that there is a k-algebra homomorphism K E → A, which is necessarily injective since it is nontrivial and K E is simple. The hypotheses imply that [A : K] ≤ |G| = [K E : k], so we must have [A : K] = [K E : k] and thus A ' K E. From this it follows that 2 × the class of A in H (G, K ) is that of (as,t).

4.2.4.7 Example: Hamilton quaternions Before going on let’s consider a simple example. The ring H of Hamilton quaternions is a 4-dimensional R-vector space with basis 1, i, j, k whose multiplication is determined by the relations

i2 = j2 = k2 = −1, ij = k = −ji. (4.2.4.7)

It is easily seen to be a division ring with center R. Let i : C → H be the R-algebra homomorphism mapping i ∈ C to i ∈ H. It is easily checked that its image is a maximal commutative subalgebra of H, and this is also clear from corollary 4.1.3.8. As usual let G = Gal(C/R), which is cyclic of order 2 with generator c (complex conjugation). On account of the relations 4.2.4.7 the group E = E(H, i) is the disjoint union C× ∪ jC×, and the projection p : E → G is p(C×) = 1, p(jC×) = c. If we choose 1 7→ 1, c 7→ j as a section of p, the corresponding 2-cocycle is

a1,1 = a1,c = ac,1 = 1, ac,c = −1. (4.2.4.8)

Conversely if we use the recipe 4.2.4.6 for the crossed product algebra and write u1 = 1, uc = j, the crossed product algebra E G is C ⊕ Cj with the multiplication determined by

2 j = −1, jz =zj, ¯ z ∈ C which is of course H.

4.2.4.8 Cyclic algebras. The cocycle 4.2.4.8 is an example of the cocycle 3.2.4.11 for a cyclic group G and an element a of a G-module A such that a ∈ AG. Recall that cocycle is ( 1 i + j < n ai,j = a i + j ≥ n. if we identify G ' Z/nZ and identify i ∈ Z/nZ with its representative in [0, n), Suppose now K/k is a cyclic Galois extension with group G. Fix as before an isomorphism G ' Z/nZ and let s ∈ G correspond to 1 ∈ Z/nZ. In the notation of 4.2.4.6, ( ui+j i + j < n uiuj = aui+j i + j ≥ n 4.2. THE BRAUER GROUP 133

s i and u1b = bu1 for b ∈ K. Then u0 is the identity of K G and ui = u1 for n 0 ≤ i < n, while u1 = au1 = a. If we write x = u1, then K E is a K-vector space with basis 1, x, . . . , xn−1 and the multiplication is determined by xb = sbx, b ∈ K, xn = a. (4.2.4.9) These algebras are (unsurprisingly) called cyclic algebras. Before the development of the cohomological approach they were one of the main tools in local class ﬁeld theory, particular when K/k was a Kummer or Artin-Schreier extension.

4.2.5 Inflation and restriction. In section 3.2.1.2 we constructed functors on categories of groups extensions that correspond to the restriction and inflation on degree two cohomology. Here we do the same things for the categories CS(K/k). Suppose first that L/k is a finite Galois extension with group G, K is an extension of k in L and H = Gal(L/K). Let (A, i) be an object in CS(L/k). By the double centralizer theorem 4.1.3.5 the centralizer CA(i(K)) is a simple k-algebra and its centralizer in A is i(K). In particular its own center is i(K), so if we make CA(i(K)) a K-algebra via the restriction i to K, CA(i(K)) is a central simple K-algebra. The equality 4.1.3.1 shows that

[CA(i(L)) : k] = [A : k]/[K : k] and then 2 2 [CA(i(L)) : K] = [A : k]/[K : k] = [L : K] 2 since [A : k] = [L : k] . Therefore i embeds L into CA(i(K)) as a maximal commutative subalgebra, and (CA(i(K)), i) is an object of CS(L/K). This construction is evidently functorial in (A, i), and we have deﬁned a functor

ResL/K : CS(L/k) → CS(L/K). (4.2.5.1)

We will also use ResL/K (A) to denote the K-algebra CA(i(K)). Denote by WA the K-vector space A where K acts on A by right multiplication via i. The action of ResL/K (A) is K-linear, whence an embedding op r : ResL/K (A) → EndK (WA). On the other hand the action of A on itself by left multiplication yields a k-linear map A → EndK (WA), which in turn yields a K-algebra homomorphism ` : K ⊗k A → EndK (WA) by adjunction. Since the image of ` lies in the centralizer of the image of r there is a K-algebra homomorphism

op (K ⊗k A) ⊗K ResL/K (A) → EndK (WA) and a quick computation shows that source and target have the same dimension. Therefore the map is an isomorphism, and ∼ K ⊗k A −→ ResL/K (A). (4.2.5.2) It follows that the the restriction functor CS(L/k) → CS(L/K) yields, on passing to isomorphism classes the map Br(L/k) → Br(L/K). 134 CHAPTER 4. THE BRAUER GROUP

4.2.5.1 Proposition For any object E of EXT(G, L×) there is a functorial isomorphism H ∼ CSL/K (ResG (E)) −→ ResL/K (CSL/k(E)) (4.2.5.3) in CS(L/K). Proof. The assertion is that for all E there is a functorial isomorphism

H ∼ L ResG (E) −→ ResL/K (L E) P compatible with the embeddings of L. Suppose a = s∈G as is an element of L E with as ∈ [L, E]s. For x ∈ K the equality xa = ax says that

X X X s x as = asx = xas s s s and if this is to hold for all x ∈ K, as 6= 0 if and only if s ∈ H. Then a = P H s∈H as can be identified with an element of L ResG E, and this identification is evidently functorial and compatible with the identifications of L as a maximal commutative subalgebra.

Suppose now that as before L/k is a finite Galois extension with group GL/k, but now K is a Galois extension of k contained in L with Galois group GK/k, and denote by π : GL/k → GK/k the canonical projection. We will construct a functor CS(K/k) → CS(L/k) correspsonding to the inflation functor for π. let (A, i) be an object of CS(K/k). As in the construction of the product ∗k we denote by VA the K-vector space A, where L acts on VA by left multiplication. op As in section 4.2.4 the homomorphism r : A → EndK (VA) defined by right multiplication induces an isomorphism

op ∼ K ⊗k A −→ EndK (VA) and tensoring with L over K yields

op ∼ L ⊗k A −→ EndL(L ⊗K VA) ⊆ Endk(L ⊗K VA)

op The centralizer of r(A ) in Endk(L⊗K VA) of the image of is a central simple k- algebra similar to A which we will denote by I(A, i), and the above isomorphism yields an embedding i : L → I(A, i) as a maximal commutative subalgebra. We G denote by InfH (A, ) the resulting object (I(A, i), i) of CS(L/k). This is clearly functorial in (A, i), so we have constructed a functor

InfL/K : CS(K/k) → CS(L/k). (4.2.5.4)

As usual we will use InfL/K (A) to denote the algebra I(A, i). Corresponding to proposition 4.2.5.1 we have:

4.2.5.2 Proposition For any object E of Ext(H,K×) there is a functorial isomorphism G CS (Inf L/k (E)) −→∼ Inf (CS (E)) (4.2.5.5) L/k GK/k L/K K/k in CS(L/k). 4.2. THE BRAUER GROUP 135

Proof. Write E = (E, i, p) and set A = K E; in the preceding construction VA is A with the K-vector space structure given by the canonical injection 0 ι : K → A. Set G = GL/k and G = GK/k; we will construct a homomorphism G G v : InfG0 (E) → InfL/K (A) as follows. Recall that elements of InfG0 (E) can be × 0 represented by pairs [x, e] with x ∈ L , e ∈ E ×G G and [xa, e] = [x, i(a)e] for × G a ∈ K . For [x, e] ∈ InfG0 (E) we deﬁne v(x, e) to be the L-linear endomorphism of L ⊗K VA given by ` ⊗ w 7→ `x ⊗ u(τ(e))w

0 where τ : E ×G G → E is the projection. Denote by v¯ the induced map

G v¯ : InfG0 (E) → Endk(L ⊗K VA).

The image of v¯ clearly lies in the centralizer of the image of the map r : L ⊗k op A → Endk(L ⊗K VA) induced by right mulitplication on A, and thus factors G through a map v : InfG0 (E) → InfL/K (A) as required. On the other hand the k-algebra homomorphism L → Endk(L ⊗K VA) embeds L into the centralizer of the image of r, yielding a k-algebra homomorphism j : L → InfL/K (A). It is easily checked that v and j satisfy the conditions of proposition 4.2.3.1. Therefore there is a k-algebra homomorphism

G f : L InfG0 (E) → InfL/K (A) compatible with the embeddings of L as a maximal commutative subalgebra of source and target. Since both source and target have the same k-dimension, f is an isomorphism. To summarize the results of this section: If L/k is a Galois extension with group G, H ⊆ G and K = LH , the restriction functor induces on isomorphism classes the homomomorphism Br(L/k) → Br(L/K) given by A 7→ K ⊗k A, and then the diagram

Br(L/K) / Ext(G, L×) / H2(G, L×) (4.2.5.6)

Res Res Br(L/K) / Ext(H,L×) / H2(H,L×) is commutative. If H is normal in G and G0 = G/H, the inﬂation functor yields on passage to isomorphism classes the canonical inclusion Br(K/k) → Br(L/k), and Br(K/k) / Ext(G0,K×) / H2(G0,K×) (4.2.5.7)

Inf Inf Br(L/k) / Ext(G, L×) / H2(G, L×) is commutative. 136 CHAPTER 4. THE BRAUER GROUP

4.3 Nonarchimedean Fields

The aim of this section is to compute the Brauer group of a discretely valued nonarchimedean field with perfect residue field, given that the Brauer group of the residue field is known. Since the case of a local field is the one of interest in class field theory we treat this first.

4.3.1 Local ﬁelds. We will need the following general result:

4.3.1.1 Proposition If K is a nonarchimedean field, any central simple K- algebra of finite degree splits over an unramified extension of K. The argument is completely parallel to that of proposition 4.1.3.11, with lemma 4.3.1.2 replaced by the following:

4.3.1.2 Lemma Suppose K is a discretely valued nonarchimedean field and D is a central division algebra over K. If every subfield of D is totally ramified over K then D = K. Proof. Suppose b ∈ D \ K and let L = K(b). By hypothesis L/K is totally ramified. Choose a uniformizer π ∈ OD. Since L/K is totally ramified we may write b = a0 + b1π with a0 ∈ K. Similarly b1 = a1 + b2π with a1 ∈ K, and by induction 2 n b = a0 + a1π + a2π + ··· + bnπ with ai ∈ K, bn ∈ L for all n. Thus b is in the closure of K(π). Since [K(π): K] is finite, K(π) is closed in D, and it follows that b ∈ K(π). Since b was any element of D, D = K(π) is commutative, so D = K. The proof of proposition 4.3.1.1 repeats that of proposition 4.1.3.11 with “separable” replaced by “unramified.” The structure of the relative Brauer group of a finite unramified extension of local fields follows from proposition 3.3.3.1 and the calculations following it.

4.3.1.3 Theorem Let L/K be a finite Galois extension of local fields with residual extension kL/k and Galois group G. Let n = [L : K] and let G ' Z/nZ be isomorphism which to 1 ∈ Z/nZ associates the lifting of the qth power Frobenius of kL to L. There is a canonical isomorphism 1 inv : Br(L/K) → / (4.3.1.1) L/K nZ Z such that 1/n mod Z is the image of the class in Br(L/K) represented by the cyclic algebra defined by the cocycle ( 1 i + j < n ai,j = (4.3.1.2) π i + j ≥ n. where π is any uniformizer of K, and as before we identify i ∈ Z/nZ with its lifting in [0, n). 4.3. NONARCHIMEDEAN FIELDS 137

Proof. This follows from the case n = 2 of proposition 3.3.3.1 and the compu- × × tation of K /NL/K (L ) in proposition 2.4.1.5. In particular every central simple algebra of finite dimension over K is a cyclic algebra. The isomorphism 4.3.1.1 is the invariant map. If A is a central simple K- algebra split by L we write invL/K (A) for invL/K ([A]) and call it the invariant of A. The discussion following proposition 3.3.3.1 shows that if invL/K (A) = r/n modulo 1, A is similar to the cyclic algebra defined by the relations xa = σax, a ∈ L, xn = πa. (4.3.1.3) If we recall that any central simple K-algebra has an unramified extension of L of K embedded as a maximal commutative subalgebra, we can give a more intrinsic description of the invariant (the classical definition, in fact). It suffices to treat the case of a central division algebra over K, and recall that for any such algebra D the absolute value of K extends uniquely to D. Denote by vD the extension to D of the normalized valuation of K. The Noether-Skolem theorem implies that any automorphism of L extends to an inner automorphism of D; in particular there is an x ∈ D such that such that xax−1 = σa for all a ∈ L, and x is determined up to a factor in L. Then

invL/K (D) = vD(x) mod Z. (4.3.1.4) In fact 4.3.1.3 shows that the x defined there can serve as the x in 4.3.1.4, and the last equality in 4.3.1.3 shows that a v (x) = = inv (D). D n L/K Suppose now L0/L is an unramified extension of degree m, so that L0/K is unramified of degree mn, and let G0 be the Galois group of L0/K. We are then in the situation of of the commutative diagram 4.2.5.7 with L0/L/K replacing L/K/k and the roles of G and G0 are reversed. The description of the inflation map in cohomology shows that

vL ∂ Br(L/K) / H2(G, L×) / H2(G, Z) o Hom(G0, Q/Z)

Inf Inf Inf

vL0 ∂ Br(L0/K) / H2(G0, (L0)×) / H2(G0, (L0)×)o Hom(G0, Q/Z) is commutative. Now G and G0 have canonical generators, the lifting to L and L0 of the Frobenius of the residual extension. If n = [L : K] and n0 = [L0 : K] = nm, this ﬁxes isomorphisms G ' Z/nZ and G0 ' Z/n0Z, as well as canonical identiﬁcations Hom(G, Q/Z) ' (1/n)Z/Z, Hom(G0, Q/Z) ' (1/n0)Z/Z making

1 Hom(G, Q/Z) / n Z/Z

Inf 0 1 Hom(G , Q/Z) / n0 Z/Z 138 CHAPTER 4. THE BRAUER GROUP where the right hand vertical arrow is the natural inclusion. Therefore

inv L/K 1 Br(L/K) / [L:K] Z/Z (4.3.1.5)

inv 0 0 L /K 1 Br(L /K) / [L0:K] Z/Z and since Br(K) is the direct limit of the relative Brauer groups Br(L/K) with L/K unramiﬁed, we have proven:

4.3.1.4 Theorem The direct limit of the invariant maps is an isomorphism

∼ invK : Br(K) −→ Q/Z. (4.3.1.6)

4.3.2 Finite ﬁelds. Along the way, the following important result got lost in the shuﬄe:

4.3.2.1 Theorem The Brauer group of a ﬁnite ﬁeld is trivial.

Proof. It suﬃces to show that Br(k0/k) is trivial for any ﬁnite extension of k. If G is the Galois group of k0/k,

0 2 0 × ˆ 0 0 × × 0 × Br(k /k) ' H (G, (k ) ) ' H (G, (k ) ) ' k /Nk0/k(k ) and the last group is trivial by lemma 2.4.1.4.

One consequence is a celebrated theorem of Wedderburn:

4.3.2.2 Corollary A ﬁnite division ring is a ﬁeld.

Proof. Such a ring is evidently central simple over its center, which is a ﬁnite ﬁeld.

Since you can’t have too many proofs of a good theorem, we include two more:

4.3.2.3 Second proof. Let D be a finite division algebra of degree d2 over its center k. A maximal subfield K of D has [K : k] = d, so all of the maximal subfields of D are isomorphic. Fix a maximal subfield K of D and let x ∈ D; then x is contained in a subfield of D isomorphic to K, and therefore conjugate to K by Noether-Skolem. Thus D is the union of conjugates of K, but a finite group can not be a union of the conjugates of a proper subgroup (exercise). Therefore D = K = k. 4.3. NONARCHIMEDEAN FIELDS 139

4.3.2.4 Third proof. Again let D be a ﬁnite division ring of degree d2 over its center k. Any k-subalgebra F ⊆ D is a division ring; then D is a left F -vector space and F is a k-vector space. For any x ∈ D let D(x) denote the centralizer of x in D. If |k| = q then |D(x)| = qd(x) for some integer d(x), and if |D| = qn then d(x)|n for all x. The class equation of D× is then

X qd(x) − 1 qn − 1 = q − 1 + q − 1 x where the sum is over representatives of the non-central conjugacy classes of D×. Suppose now n > 1 and let P (X) be the nth cyclotomic polynomial, i.e. Q P (X) = ζ (X − ζ) where ζ runs through the nth roots of 1 in C. Then P (X) ∈ Z[X] (another exercise) and for any m < n dividing n, P (X) divides (Xn − 1)/(Xm − 1). Then P (q) divides qn − 1 and all of the terms (qd(x) − 1)/(q − 1) in the above sum. Therefore P (q) divides q − 1, but this is impossible Q since P (q) = ζ (q − ζ) and |q − ζ| > q − 1 for all primitive nth roots ζ of 1. We can now complete our discussion of the structure of division algebras over a local ﬁeld.

4.3.2.5 Theorem Suppose D is a division algebra of finite degree n2 over a local field K. Then D has order in Br(K) and its invariant is a/n with gcd(a, n) = 1. The residue ring of D is a finite field, and

eD/K = fD/K = n. Conversely if A is a central simple K-algebra of degree n2, invariant a/n and gcd(a, n) = 1, A is a division algebra.

2 Proof. We know that we must have eD/K fD/K = [D : K] = n , and fL/K ≥ n, so we must show eD/K ≥ n. The proof of the equality 4.3.1.4 showed that there is an x ∈ D such that xn = πa where π is a uniformizer of K, where a/n = invK (D). It thus suﬃces to show that a and n are relatively prime, which will complete the proof. We can assume that 0 ≤ a < n. Suppose d is a common divisor of a and n, choose x ∈ D satisfying xn = πa and set y = xn/d and τ = πa/d. Then yd − τ d = xn − πa = 0 while on the other hand yd − τ d = (y − τ)(yd−1 + yd−2τ + ··· + td) = 0 and thus y = τ or yd−1 + yd−2τ + ··· + τ d = 0, but neither of these is possible if d > 1 since 1, x, . . . , xn−1 are linearly independent over K. For the last assertion it suﬃces to observe that we must have A ' Mm(D) for some central K-division algebra D, and then invK (D) = a/n. Since a and n are relatively prime, [D : K] = n2 and A = D. 140 CHAPTER 4. THE BRAUER GROUP

4.3.3 The General Case. Suppose now K is any discretely valued nonarchimedean field with perfect residue field and L/K is a finite unramified Galois extension with group G, normalized valuation vL and residual extension kl/k. The exact sequence × × vL 1 → OL → L −−→ Z → 0 (4.3.3.1) is split exact as a sequence of G-modules (since L/K is unramified, a uniformizer π of K is also a uniformizer of L, so it suffices to lift 1 ∈ Z to a uniformizer of K). The long exact sequence of cohomology then decomposes into split short exact sequences

n × n × n 0 → H (G, OL ) → H (G, L ) → H (G, Z) → 0. (4.3.3.2)

Recall that a ﬁltration (An)n≥0 of an abelian group A is complete and separated if the natural homomorphism

A → lim A/A ←− n n is an isomorphism.

4.3.3.1 Lemma Suppose A is a G-module and (An)n∈N is a ﬁltration of A for n which A is complete and separated. If H (G, An/An+1) = 0 for all n ≥ 0 then Hn(G, A) = 0.

Proof. Suppose u is an n-cocycle for (G, A). The exact sequences

n n n H (G, An+1) → H (G, An) → H (G, An/An+1) = 0 show that we can deﬁne a sequences of n-cocycles (an) and n-cochains (bn) for (G, An) such that a = a0 and

an = ∂bn + an+1 for all n ≥ 0. Since A is complete and separated for the filtration, the sum P b = n bn converges and the above equalities show that a = ∂b, so the class of a in Hn(G, A) is zero. n × n n Recall the filtration U ⊆ OL where x ∈ U if and only if x ≡ 1 mod m , 0 1 × n n+1 and that U /U ' kL and U /U ' kL as G-modules. By proposition 3.4.1.1, m n n+1 n H (G, U /U ) ' H (G, kL) = 0 for n > 0 since G is also the Galois group of kL/k. The previous lemma implies that Hn(G, U 1) = 0 n > 0 and finally the long exact sequence obtained from

1 × × 1 → U → OL → kL → 1 4.3. NONARCHIMEDEAN FIELDS 141 yields isomorphisms n × ∼ n × H (G, OL ) −→ H (G, kL ) (4.3.3.3) for all n > 0. From 4.3.3.2 we get split short exact sequences n × n × n 0 → H (G, kL ) → H (G, L ) → H (G, Z) → 0 (4.3.3.4) for n > 0. If we set n = 2 and recall the isomorphism 2 H (G, Z) ' Hom(G, Q/Z) the result is an exact sequence

0 → Br(kL/k) → Br(L/K) → Hom(GL/K , Q/Z) → 0 where we have replace G by GL/K . Passing to the direct limit over unramiﬁed extensions L/K then yields:

4.3.3.2 Theorem If L/K is an unramified finite Galois extension of discretely valued nonarchimedean fields with residual extension kL/k. If k is perfect and G = Gal(L/K) there is a split exact sequence

0 → Br(k) → Br(K) → Homcont (G, Q/Z) → 0 (4.3.3.5) where Homcont (G, Q/Z) denotes the group of continuous homomorphisms G → Q/Z for the discrete topology of Q/Z.

Proof. The only point needing explanation is that Homcont (G, Q/Z) is the direct limit of the groups Hom(GL/K , Q/Z). In fact the individual Hom(GL/K , Q/Z) inject into the direct limit and induce continuous homomorphisms G → Q/Z. On the other hand since Q/Z is discrete any continuous homomorphism G → Q/Z factors through a finite quotient. The splitting is not canonical, but is determined by the choice of a uniformizer of K. The theorem shows a central simple K-algebra is similar to the product of algebras of one of two kinds, which we will now analyze. The algebras arising from the term Hom(G, Q/Z) in 4.3.3.5 are cyclic, since any homomorphism G → Q/Z with finite image factors through a finite cyclic group. Formally they look like the algebras defined by 4.3.1.3, but now σ can be any generator of a cyclic quotient of G (and n is the order of the quotient). The algebras corresponding to the term Br(k) are more interesting. Let us re- 2 × 2 × call how they arise: they belong to the subgroup H (GL/K , OL ⊂ H (GL/K ,L ) of Br(L/K) for some unramified L/K. In other words there is a cocycle repre- × senting the Brauer class that takes its values in OL ; following the usual terminology of algebraic geometry we could say that these classes have “good reduction.” 2 × If as,t ∈ Z (G, OL ) is any such cocycle, its reduction a¯s,t modulo the maximal 2 × ideal represents a class in Br(kL/k); conversely if a cocycle a¯s,t ∈ Z (G, kL is 2 × given its “Teichmüller lift” [¯as,t] ∈ Z (G, OL ) yields an element of Br(L/K). In fact we can say more. Let R be a local ring with maximal ideal m; an R-algebra A is an Azumaya algebra if it is free of finite type as an R-module, op and the natural map A ⊗R A → EndR(A) is an isomorphism (the map is the same one as in 4.1.2.9). 142 CHAPTER 4. THE BRAUER GROUP Chapter 5

The Reciprocity Isomorphism

Let L/K be a Galois extension of local ﬁelds with group G. In this chapter we ab ∼ × × establish an isomorphism G −→ K /NL/K L , which for historical reasons is called the reciprocity map. Our method uses an explicit determination of the relative Brauer group Br(L/K), which so far has been carried out only if L/K is unramiﬁed.

5.1 F -isocrystals

The calculation of the Brauer group of a local field in the last section is nonconstructive, since the proof of existence of an unramified splitting field was nonconstructive. In this section we realize central simple algebras of a local field as endomorphism algebras of simple modules over a noncommutative ring, the Dieudonné ring. In this section K is a discretely valuded nonarchimedean field with residue field k of characteristic p > 0, q = pf is a power of p and σ : K → K is a lifting of the qth power Frobenius. Further restrictions will be made on the way.

5.1.1 Deﬁnitions. An F -isocrystal is a ﬁnite-dimensional K-vector space V endowed with an σ-linear isomorphism F : V → V , i.e. an additive map satisfying F (av) = σaF (v) for all a ∈ K. The map F is called the Frobenius structure of the F -isocrystals. A morphism f :(V,F ) → (V 0,F 0) is a K-linear map f : V → V 0 such that f V / V

F F 0 V / V 0 f

We denote by FIsoc(K, σ) the category of F -isocrystals on K. It is an abelian category: the kernel and cokernel of a morphism of F -isocrystals are F -isocrystals

143 144 CHAPTER 5. THE RECIPROCITY ISOMORPHISM and behave in the expected way. If σ is understood we will write FIsoc(K) for FIsoc(K, σ). Certain basic constructions of linear algebra can be made for F -isocrystals, which we describe in the next few paragraphs. The direct sum of F -isocrystals (V,FV ), (W, FW ) is V ⊕ W with the Frobenius structure FV ⊕ FW . The tensor product V ⊗ W of F -isocrystals (V,FV ) and (W, FW ) is V ⊗K W with the Frobenius structure F (x ⊗ y) = FV (x) ⊗ FV (y) (note that this is well-defined). Similarly for exterior and symmetric products. Suppose L/K is a finite extension and σ : K → K extends to a lifting σ0 : L → L of the q-th power Frobenius of the residue field of L. The extension of scalars to L of an F -isocrystal (V,F ) is the L-vector space L ⊗K M with the Frobenius structure F (a ⊗ v) = σ0(a) ⊗ F (v). This yields an exact functor

K⊗ : FIsoc(K) → FIsoc(L).

Extension of scalars does not change the rank. In the same situation as before we can deﬁne the restriction of scalars to K of an F -isocrystal (V,F ) on L: it is simply V viewed as K-vector space endowed with the same F . This deﬁnes an exact functor

ResL/K : FIsoc(L) → FIsoc(K) and then dimK ResL/K (V ) = [L : K] dimL V. For any ﬁnite extension L/K there is a functorial isomorphism

∼ HomK (M, ResL/K (N)) −→ HomL(L ⊗ M,N) (5.1.1.1) induced by the usual adjunction morphism. The nth iterate of a σ-F -isocrystal (M,F ) is the σn-F -isocrystal V endowed with F n : M → M. The Tate twist VK (n) for n ∈ Z is the F -isocrystal K with the Frobenius structure F (a) = πnaσ. More generally, if V is any F -isocrystal, the nth twist of V is the tensor product V ⊗ V (n).

5.1.2 The Dieudonné ring Our ﬁrst task is to classify F -isocrystals on K when k is algebraically closed. The main tool is the Dieudonné ring, which we deﬁne as the crossed product algebra

DK = K Z (5.1.2.1) where Z acts on K by having 1 ∈ Z act as σ. In the notation of section 4.2.3 the additive and multiplicative identites of DK are [0, 0]0 and [1, 0]0 since additive notation is used for Z! We use the canonical embedding K,→ K Z to identify 5.1. F -ISOCRYSTALS 145

K with a subring of DK , so if in the notation of section we set F = [1, 1]1 then any a ∈ DK has a unique expression

X n a = anF (5.1.2.2) n∈Z and multiplication in DK is done using the formula

F a = aσF (5.1.2.3) for a ∈ K. If a ∈ DK is given by 5.1.2.2 and a 6= 0 let r (resp. s) is the largest (resp. smallest) integer n such that an 6= 0 we deﬁne the degree of a to be

deg(a) = r − s. (5.1.2.4)

It is immediate that

deg(a) ≥ 0 (5.1.2.5) deg(ab) = deg(a) + deg(b) (5.1.2.6) for all nonzero a, b ∈ DK . If we deﬁne deg(0) = −∞ the second formula holds for all a and b in DK . From this we see that DK has no left or right divisors of n zero, and the units of DK are the elements of degree zero, i.e. monomials aF for a ∈ K and n ∈ Z. Any nonzero element of DK is a unit times a “polynomial in F ”, so the usual division algorithm holds in the following form:

0 0 5.1.2.1 Lemma For any a, b ∈ DK with b 6= 0 there are q, q , r and r ∈ DK such that

a = bq + r = q0b + r0 deg(r), deg(r0) < deg b. (5.1.2.7)

The usual argument then yields:

5.1.2.2 Proposition Any left or right ideal in DK is principal.

A left DK -module M has a K-vector space structure by the restriction of scalars K → DK , and the map F : M → M given by left multiplication by F ∈ DK is semilinear by 5.1.2.3, so (M,F ) is an F -isocrystal on K. Conversely an F -isocrystal (V,F ) has a natural structure as a left DK -module, and this allows us to identify FIsoc(K) with the category of left DK -modules of finite dimension over K. For the rest of this section we assume that the residue fields k of K is algebraically closed (we will frequently repeat this hypothesis for emphasis). By corollary 1.3.6.3 K has a uniformizer π that is fixed by σ. We fix such a uniformizer; from 5.1.2.3 it follows that π is in the center of DK . 146 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

It will be useful to adopt the convention that rational numbers, unless speci- ﬁed otherwise, are written in lowest terms. The integer 0 in lowest terms is 0/1. With this convention we deﬁne, for any λ = r/d ∈ Q, VK (λ) is the F -isocrystal

d r VK (λ) = DK /DK (F − π ) (5.1.2.8) and note that when λ = n/1 with n ∈ Z this coincides with our earlier deﬁnition. The division algorithm 5.1.2.7 shows that VK (λ) has dimension d; in fact the d−1 images of 1, F,...,F ∈ DK are a K-basis of VK (λ). It will sometimes be useful to write this basis as

λ i d r ei = F mod (F − π ) 0 ≤ i < d. (5.1.2.9)

d With λ = r/d as before we define Kλ to be the fixed field of σ . When the σ residue field k is algebraically closed Kλ is the unramified extension of K0 = K of degree d, and in general is some subfield of this containing K0. In any case it is a local field by theorem 1.3.6.5. The definition 5.1.2.8 shows that right multiplication by elements of Kλ defines an embedding Kλ → Dλ. The goal of the next few sections is to show that the VK (λ) are simple DK - modules, and that any F -isocrystal on K is a direct sum of modules of this form.

5.1.3 Cyclic Modules. The ﬁrst task is to compute the endomorphism rings of the V (λ) and the Ext groups Exti (V (λ),V (µ)). We may as well K DK K K look at the problem for cyclic modules in general, so in this section R is any ring with identity, f ∈ R is not a right zero-divisor, and M is the left R-module R/Rf. Then ·f 0 → R −→ R → M → 0 is a resolution of M by free R-modules, and for any left R-module N the groups i ExtR(M,N) are computed by the complex

f· · · · → 0 → N −→ N → 0 → · · · whose nonzero terms are in degrees 0 and 1. Denoting by f N the kernel of left multiplicatin by f in N, we ﬁnd that

HomR(M,N) ' f N, 1 ExtR(M,N) ' N/fN, (5.1.3.1) n ExtR(M,N) = 0 n ≥ 2.

r d We now apply this calculation to R = DK and f = F − π :

5.1.3.1 Proposition Suppose k is algebraically closed and λ, µ ∈ Q.

(i). If λ 6= µ then HomDK (VK (λ),VK (µ)) = 0. 5.1. F -ISOCRYSTALS 147

(ii). Extn (V (λ),V (µ)) = 0 for all λ, µ ∈ and n ≥ 1. DK K K Q Proof. Write λ = r/d as before. By the isomorphisms 5.1.3.1 it suﬃces to show d r that left multiplication by F − π on VK (µ) is (1) injective if µ 6= λ, and (2) surjective in any case. Write µ = r0/d0. d r (1) Suppose x is in the kernel of left multiplication by F − π on VK (µ). d0−1 µ µ Recall that 1,F,...,F ∈ DK reduce to a K-basis of e0 , . . . , ed0−1 of VK (µ); P µ then with x = i aiei ,

0 dd0 X σdd dr0 µ rd0 X rd0 µ F (x) = ai π ei and π x = π aiei . i i

0 0 dd0 rd0 Since r d 6= rd , we cannot have F (x) = π x unless all ai are zero. (2) Again with µ = r0/d0, the identity

0 0 0 0 0 F dd − πrd = (F d − πr)(F d(d −1) + πrF d(d −2) + ··· + πr(d −1))

dd0 rd0 P µ shows that it suﬃces to prove that F − π is surjective. For x = i aiei ∈ VK (µ) we see that

0 dd0 rd0 X σdd rd0 µ (F − π )x = (ai − π ai)ei i the assertion follows from corollary 1.3.6.4

5.1.4 The Endomorphism Ring of VK (λ). When N = M the ﬁrst isomorphism of 5.1.3.1 is End(M) ' f M and it can be described explicitly as follows: α ∈ End(M) corresponds to right multiplication by any element of R lifting α(e), where e ∈ M = R/Rf is the image of 1 ∈ R. It is easy to check directly that this is well-deﬁned: if m ∈ M and α ∈ f M, we choose m ∈ R and a representing m and α(e) respectively; the image of ma in M is independent of the choice of m since any other choices would be m + xf, a + yf with x, y ∈ R, and then

(m + xf)(a + yf) = m a + element of Rf since fa = 0 in M. We can now determine the structure of the endomorphism ring

Dλ = End(VK (λ))

d r which we identify, as in 5.1.3, with the kernel of F −π on VK (λ). As before the i λ image of F ∈ DK in VK (λ) for 0 ≤ i < d will be denoted by ei with i ∈ Z/dZ, λ λ d r and e0 , . . . , ed−1 is a K-basis of VK (λ). The kernel of F − π is the set of d−1 a0 + a1F + ··· + ad−1F with ai ∈ Kλ, and recall that if k is algebraically closed Kλ is an extension of K0 of degree d. 148 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

As before we identify a ∈ Kλ with the corresponding endomorphism (right multiplication by a) of VK (λ), and denote by ξ the endomorphism corresponding to right multiplication by F . From this it is clear that Dλ is a Kλ-vector space with basis 1, ξ, . . . , ξd−1, and that

ξd = πr, aξ = ξaσ (5.1.4.1) in Dλ.

5.1.4.1 Proposition Suppose the residue ﬁeld of K is algebraically closed. If σ λ = r/d in lowest terms, Dλ is a central division algebra over K0 = K of 2 degree d , invariant −λ modulo Z and containing Kλ as a maximal commutative subﬁeld.

Proof. From 5.1.4.1 we see that the center of Dλ is K0. Since π is a unit in Dλ so is x, and if we set x = ξ−1 the relations

xd = π−r xa = σax show that D is a central simple Kσ-algebra with invariant −r/d. Since r and d are relatively prime, D is a division algebra by theorem 4.3.2.5. The last statement is clear.

5.1.5 The classiﬁcation: k algebraically closed. We now suppose that the residue ﬁeld of K is algebraically closed. Let λ = r/d with the fraction in lowest terms, let L = K(π1/d), and extend σ to L by setting σ(π1/d) = π1/d. 1/d i/d dλ Then OL = OK [π ] has the OK -basis π for 0 ≤ i < d. Since e0 ∈ VL(dλ) i/d dλ is a basis of the rank one F -isocrystal VL(dλ), the π ei for 0 ≤ i < d form a basis of the F -isocrystal ResL/K VL(dλ). It is easily checked that the linear i/d dλ λ map sending π e0 to ei is an isomorphism ∼ ResL/K VL(dλ) −→ VK (λ). (5.1.5.1)

n n−1 5.1.5.1 Lemma Let F + a1F + ··· + an−1F + an be an element of DK such that ai 6= 0 for some i. Suppose v (a ) r inf π i = λ = gcd(r, d) = 1 (5.1.5.2) i i d

−d × and let L = K(π . There are b1, . . . , bn−1 ∈ L and u ∈ OL such that

n n−1 n−1 n−2 λ (F + a1F + ··· + an−1F + an)u = (b0F + b1F + ··· + bn−1)(F − π ) (5.1.5.3) in DL.

Proof. Since π is in the center of DK there is an automorphism of DK such that F 7→ πλF . Applying this automorphism to 5.1.5.3 and dividing by πλ yields λ −λi −λi an equality similar to 5.1.5.3 with ai, bi and F − π replaced by π ai, π bi 5.1. F -ISOCRYSTALS 149 and F − 1 respectively. Thus it suﬃces to prove the lemma in the case where all ai are in OK , at least one is a unit, and λ = 0. The bi are determined by n n−1 n−1 n−2 (F + a1F + ··· + an−1F + an)u = (b0F + b1F + ··· + bn−1)(F − 1) (5.1.5.4) or equivalently

σn n σn−1 n−1 n n−1 u F +u a1F +···+uan = b0F +(b1−b0)F +···+(bn−1−bn−2)F −bn so that u and the bi satisfy σn u = b0 σn−1 a1u = b1 − b0 ······ σ an−1u = bn−1 − bn−2

an = −bn−1. If these equations are satisﬁed, then u is a solution of

σn σn−1 u + a1u + ··· + anu = 0 (5.1.5.5) and conversely if u is a solution of 5.1.5.5 then bi ∈ OL can be found satisfying 5.1.5.4. Finally, since k is algebraically closed, 5.1.5.5 always has a solution by lemma 1.3.6.2.

5.1.5.2 Lemma If k is algebraically closed and M is a nonzero F -isocrystal, there is a nonzero homomorphism M → VK (λ) for some λ. Proof. Set L = K[π1/d] as before. By replacing M by a quotient, if necessary, we can assume M is simple, and therefore cyclic, say M = DK /DK P for some n n−1 P ∈ DK . Write P = F + a1F + ··· + an; since F is bijective on M we must λ have an 6= 0, and the previous lemma is applicable. Write P u = Q(F −π ) with λ = r/d as in 5.1.5.2 and Q ∈ DL. Since M ' DK /(P u), we see that L⊗K M has λ VL(dλ) ' DL/(F −π ) as a quotient, say via a map L⊗K M → VL(dλ). Applying the restriction functor and using 5.1.5.1 yields a morphism M → VK (λ) which is nonzero, for if it were zero the morphism L ⊗K M → VL(dλ) corresponding to it by adjunction would also be zero.

5.1.5.3 Proposition The F -isocrystal VK (λ) is a simple object of FIsoc(K).

Proof. Suppose M is a proper submodule of VK (λ). By lemma 5.1.5.2 there is a nontrivial morphism f : VK (λ)/M → VK (µ) for some µ. By the ﬁrst part of proposition 5.1.3.1 we must have λ = µ. The composite map

f VK (λ) → VK (λ)/M −→ VK (λ) is an element of the division algebra End(VK (λ)) = Dλ, and is therefore an isomorphism. It follows that M = 0. 150 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

5.1.5.4 Proposition If k is algebraically closed, a simple F -isocrystal is isomorphic to some K(λ). For λ 6= µ the isocrystals VK (λ) and VK (µ) are nonisomorphic.

Proof. For any M there is a surjective morphism M → VK (λ) for some λ which, if M is simple, must be an isomorphism. The second statement follows from the ﬁrst part of proposition 5.1.3.1. We can now prove Dieudonné’s classiﬁcation theorem [6], c.f. also Manin [9]:

5.1.5.5 Theorem If k is algebraically closed, any F -isocrystal on K is a direct sum of simple F -isocrystals, and a simple F -isocrystal is isomorphic to VK (λ) for a unique λ ∈ Q.

Proof. We have shown that every VK (λ) is simple (proposition 5.1.5.4), and proposition 5.1.3.1 shows that VK (λ) and VK (µ) are nonisomorphic when λ 6= µ. This establishes the last statement. Furthermore lemma 5.1.5.2 shows that any nonzero F -isocrystal has some VK (λ) as a quotient: since VK (λ) is simple, a nonzero homomorphism V → VK (λ) must be surjective. To conclude the proof 1 it suﬃces to show that Ext (V,VK (λ)) = 0 for any F -isocrystal V , but this follows by induction on dim V from the equality Ext1 (V (µ),V (λ)) = 0 K DK K K (proposition 5.1.3.1).

5.1.6 Slopes. Theorem 5.1.5.5 asserts that an F -isocrystal M over a ﬁeld K with algebraically closed residue ﬁeld k has a direct sum decomposition

M mλ M = VK (λ) (5.1.6.1) λ∈Q and the set of λ ∈ Q for which mλ 6= 0 are the slopes of M. For each λ ∈ S we deﬁne the multiplicity of λ in M to be

multλ(M) = mλ dimK VK (λ). (5.1.6.2)

Explicitly, if λ = r/d then multλ(M) = dmλ. We may rewrite 5.1.6.1 as M M = M λ (5.1.6.3) λ∈S where S is a ﬁnite subset of Q and M λ is the sum of all sub-F -isocrystals of M isomorphic to K(λ). This decomposition is functorial in M: if N is another λ F -isocrystal on K with slope decomposition N = ⊕λN , then any morphism F : M → N induces morphisms M λ → N λ for all λ; this follows immediately from part (i) of proposition 5.1.3.1. We say M is isopentic of slope λ if multµ = 0 for all µ 6= λ (some authors use the term isoclinic). We say that M is unit-root if it is isopentic of slope 0. 5.1. F -ISOCRYSTALS 151

It will be useful to write the slopes of an F -isocrystal as an increasing sequence λ1 ≤ λ2 ≤ · · · ≤ λd (5.1.6.4) in which each slope is repeated as many times as its mulitplicity, so that d is the dimension of M. Still supposing that the residue ﬁeld of K is algebraically closed, let (V,F ) is an F -isocrystal on K with slopes λ1 ≤ λ2 ≤ · · · ≤ λd. It is clear from the deﬁnitions that

n The n-th iterate (V,F ) has slopes nλ1 ≤ nλ2 ≤ · · · ≤ nλd (5.1.6.5)

The twist M ⊗ K(i) has slopes λ1 + i ≤ λ2 + i ≤ · · · ≤ λd + i. (5.1.6.6)

For more complicated operations we need the following description of the slopes. If K is any nonarchimedean field, a lattice in a K-vector space V of finite dimension is a finite OK -submodule M ⊂ V such that the natural map

K ⊗OK M → V is an isomorphism.

5.1.6.1 Proposition Suppose the residue ﬁeld of K is algebraically closed. The slopes of an F -isocrystal (M,F ) are contained in the interval [λ, µ] if and only if for some uniformizer π of K and some lattice M0 ⊂ M there are constants C, D such that [nµ]+C n [nλ]+D π M0 ⊆ F (M0) ⊆ π M0 (5.1.6.7) independently of n. Proof. We ﬁrst observe that the validity of 5.1.6.7 is independent of π, and it is also independent of the choice of M0. In fact if 5.1.6.7 holds for M0 and M1 is any another lattice there are integers a and b such that

a b π M1 ⊆ M0 ⊆ π M1 and then 5.1.6.7 holds for M1 with C and D replaced by C + a − b and D + b − a respectively. To show that the condition 5.1.6.7 is necessary it therefore suﬃces to show that it holds for M = VK (λ) with µ = λ, for we may take as the lattice the direct sum of lattices in each summand of M of the form V (λ). If we take 0 λ the lattice to be the OK -span VK (λ) of the standard basis {ei } of VK (λ) and λ = r/d, then m [mλ] F e0 = π em where as before we view the subscripts as elements of Z/dZ. This leads to the estimate m [mλ]+C F ei = π ei+m (5.1.6.8) with C = 0 or r depending on i and m, and with the same convention on subscripts. Then 5.1.6.7 with µ = λ follows immediately. Suppose conversely that ν is a slope of M and 5.1.6.7 holds. Choose a direct summand of M isomorphic to VK (ν), let Vν be a lattice in this summand and 152 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

0 0 let V be an F -stable sub-OK -module of M such that Vν ⊕ V is a lattice in M. By 5.1.6.7 we have

[nµ]+C n [nλ]+D π Vν ⊆ F (Vν ) ⊆ π Vν and by our previous result,

[nν]+E n [nν]+G π Vν ⊆ F (Vν ) ⊆ π Vν for all n and some constants E, G, from which it follows that λ ≤ ν ≤ µ. From this we see how slopes vary with base change:

5.1.6.2 Corollary Suppose L is an extension of K with ramiﬁcation index e. (1) If V is an F -isocrystal on K with slopes

λ1 ≤ λ2 ≤ · · · ≤ λd the slopes of L ⊗K V are

eλ1 ≤ eλ2 ≤ · · · ≤ eλd. (2) If V is an F -isocrystal on L with slopes

λ1 ≤ λ2 ≤ · · · ≤ λd the slopes of ResL/K V are

λ1/e ≤ λ2/e ≤ · · · ≤ λd/e. Proof. The first assertion is clear from the definitions. For the second it suffices to check it for V = VK (λ), which is done in the same way as in the argument for 5.1.5.1. Proposition 5.1.6.1 shows that the slopes of an F -isocrystal behave like the p-adic ordinals of the eigenvalues of a linear transformation with respect to the operations of linear algebra. For more precise statements see exercises 5.1.7.4 and 5.1.7.5.

5.1.7 Exercises

5.1.7.1 Let V be a K-vector space. Show that a σ-linear map F : V → V is (σ) (σ) the same as a linear map V → V , where V = K ⊗K,σ V .

5.1.7.2 Let V and W be F -isocrystals on K. If f ∈ HomK (V,W ), show that −1 F f V −−−→V V −→ W −−→FW W. is an element of HomK (V,W ) as well, and that the above map deﬁnes a Frobe- nius structure on Hom(V,W ). It is called the internal Hom of V and W , and usually denoted by Hom(V,W ). 5.1. F -ISOCRYSTALS 153

5.1.7.3 The F -isocrystal V ∨ = Hom(V,V (0)) is the dual of V . Show that the internal Hom of two F -isocrystals V , W is

Hom(V,W ) ' W ⊗ V ∨. (5.1.7.1)

Show also that there is a canonical isomorphism

∼ Hom(VK (0), Hom(V,W )) −→ Hom(V,W ) (5.1.7.2) expressing the fact that the elements of Hom(V,W ) are the elements of Hom(V,W ) ﬁxed by the Frobenius structure of Hom(V,W ).

5.1.7.4 Let K0 be a local field whose residue field has cardinality q, V0 a K0-vector space of finite dimension, and suppose F0 : V0 → V0 is a linear endomorphism of V0. Let K be the completion of a maximal unramified extension of

K0 and set V = K ⊗K0 V0. Denote by σ the lifting to K of the qth power Frobe- nius of its residue ﬁeld, so that σ|K0 is the identity. Finally suppose F : V → V is the extension of scalars

F (a ⊗ v) = σ(a) ⊗ F0(v) of F0. Show that (V,F ) is an F -isocrystal on V whose slopes are the p-adic alg ordinals of the eigenvalues of F0 in K0 (use proposition 5.1.6.1) (reduce to the case where the eigenvalues of F0 lie in K0, and use Jordan normal form.

5.1.7.5 Here is another application of proposition 5.1.6.1. Suppose V , W are F -isocrystals with slopes λ1, . . . , λm and µ1, . . . , µn respectively. then:

• the slopes of V ⊕ W are λ1, . . . , λm, µ1, . . . , µn;

• the slopes of V ⊗ W are the λi + µj for all i, j;

• the slopes of Hom(V,W ) are the −λi + µj for all i,

∨ • the slopes of the dual V are the −λi, and

r • the slopes of the exterior power ∧ V are the λi1 + λi2 + ··· + λir for all sequences of integers i1 < i2 < ··· < ir. In particular the highest exterior power det(V ) has the single slope λ1 + λ1 + ··· + λn, where n is the rank of V

(the listing of slopes in the above is not necessarily in order of magnitude). Deduce from these assertions that if k is algebraically closed and λ, µ ∈ Q,

d VK (λ) ⊗ VK (µ) ' VK (λ + µ) and determine d in terms of λ and µ. 154 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

5.2 The Fundamental Class

For the rest of this chapter K is a local field with residue field k. As in the last section Knr is the completion of a maximal unramified extension of K. We fix a uniformizer π of K.

nr nr 5.2.1 Some K -algebras. Rings of the form K ⊗K A for some K-algebra A will be so frequent in what follows that we denote this construct by

nr AK = K ⊗K A (5.2.1.1) which is functorial in A and Knr . Thus when L/K is a Galois extension the nr Galois group Gal(L/K) acts on LK with invariants KK ' K . The map σ ⊗ 1 : LK → LK induced by σ will frequently be abbreviated by σ, and its σ subring of invariants is (AK ) = A. When L/K is a Galois extension the actions of Gal(L/K) and σ on LK commute. Let L/K be a ﬁnite separable extension of K of degree d and residual degree nr f. Then LK is a direct sum of f copies of the completion L of a maximal unramiﬁed extension of L. In fact we can identify LK with a direct sum of copies of Lnr in a canonical way by choosing an embedding Knr ,→ Lnr such that Knr / Lnr (5.2.1.2)

K / L commutes. Then L and Knr are identiﬁed with subﬁelds of Lnr , and the map

∼ nr f −i LK −→ (L ) x ⊗ y 7→ (σ (x)y)0≤i

−1 −(f−1) a(x0, x1 . . . , xf−1) = (ax0, σ (a)x1, . . . , σ (a)xf−1) (5.2.1.4) and the natural embedding L → LK is

a 7→ (a, a, . . . , a). (5.2.1.5)

f Then σL|K = σ where σL is the lifting of Frobenius to L, and σ : LK → LK is

(x0, x1, . . . , xf−1) 7→ (σL(xf−1), x0, x1, . . . , xf−2) (5.2.1.6) with respect to 5.2.1.3. It is easily checked that a choice of commutative diagram 5.2.1.2 amounts to nr a choice of x ∈ Spec(LK ) and an identiﬁcation κ(x) ' L : here x is the kernel nr of the morphism LK → L canonically determined by 5.2.1.2, and corresponds 5.2. THE FUNDAMENTAL CLASS 155 to the composition of 5.2.1.3 with the projection onto the i = 0 summand; the other projections are this one composed with powers of σ. Finally, it is clear from 5.2.1.3 could be written more naturally as an isomorphism

∼ M i ∗ nr LK −→ (σ ) L (5.2.1.7) 0≤i

Lnr −→∼ (σi)∗Lnr x 7→ 1 ⊗ x.

When L/K is unramified we may identify Knr ' Lnr canonically, and the diagram 5.2.1.2 identifies L with a subfield of Knr . The arithmetic Frobenius σarith = 1 ⊗ (σ|L) is a generator of Gal(L/K) and for the decomposition 5.2.1.3 is given by

σ σ σ σ σarith (a0, a1, . . . , af−1) = ( a1, a2,..., af−1, a0). (5.2.1.8)

× × The ﬁeld norm NL/K : L → K extends to a homomorphism

× nr × NL/K : LK → (K ) (5.2.1.9)

nr × which is the norm for the ring extension K → LK . If x ∈ LK corresponds (xi) under the decomposition 5.2.1.3, NL/K (a) is given by

Y i NL/K (x) = σ (NLnr /Knr (xi)). (5.2.1.10) 0≤i

× We denote by wL/K : LK → Q the homomorphism

−1 wL/K (x) = [L : K] v(NL/K (x)) (5.2.1.11)

× where the numerical factor guarantees that wL/K extends the valuation of K ⊂ × nr nr LK . If vL is the unique valuation of L extending those of L and K ,

X −1 wL/K (x) = f vL(xi). (5.2.1.12) 0≤i

× 5.2.2 “Lubin-Tate” F -isocrystals. If α ∈ LK ,

σ Fα : LK → LK Fα(x) = α · x (5.2.2.1) deﬁnes a Frobenius structure on LK , and we denote by LK (α) the F -isocrystal (LK ,Fα). We denote by D(α) the endomorphism ring of LK (α); it is a central simple K-algebra of degree d2, where as before d = [L : K]. Since σ ⊗ 1 is L-linear, right multiplication deﬁnes an embedding L → D(α). 156 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

× 5.2.2.1 Proposition For any finite separable extension L/K and α ∈ LK , the F -isocrystal LK (α) is isopentic of slope wL/K (α). Proof. The observation 5.1.6.5 shows that it suffices to prove that the dth iterate of LK (α) has slope dwL/K (α) for any fixed d. We will take d = [L : K]. × If α ∈ LK and x ∈ LK decompose as α = (αi) and x = (xi) according to 5.2.1.3, the Frobenius structure Fα is

σL Fα(xi) = (α0 · xf−1, α1x0, . . . , αf−1xf−2) by 5.2.1.6. Iterating f times yields

f σL Fα (xi) = (βi · xi)

f where σL = σ as before, and

σL βi = α0α1 ··· αi · (αi+1 ··· αf−1)

nr Since σL does not change the valuation of an element of L , all of the βi have the same valuation, namely X vL(βi) = vL(αj). 0≤j

Set e = eL/K ; then d = ef and

e−1 e d e σL 1+σL+···+σ σ F (xi) = F (βi · xi) = ( L βi L xi) and note that

1+σ +···+σe−1 X 1 v ( L L β ) = ev (β ) = ef v (α ) = dw (α). L i L i f L j L/K 0≤j

f nr f Let L0 ⊂ LK be the lattice corresponding to (OLnr ) ⊂ (L ) . Since dwL/K (α) ∈ Z, the last equality says that

d dwL/K (α) F (L0) = π L0

nd ndw for any uniformizer π of K. Then F (L0) = π L/K L0, and proposition d 5.1.6.1 shows that (LK ,Fα) has slope dwL/K , as required.

5.2.2.2 The unramiﬁed case. Let’s consider the case when L/K unramiﬁed of degree d = f. Pick a r ∈ Z that is relatively prime to d and set α = (πr, 1,..., 1) (5.2.2.2)

nr f nr d where as above we identify LK ' (L ) = (K ) . By the proposition LK (α) has slope r/d, and as it has rank d it must be isomorphic to VK (r/d). In fact this is easy to write down an explicit isomorphism from explicit formula for Fα in the proof of proposition 5.2.2.1. 5.2. THE FUNDAMENTAL CLASS 157

5.2.2.3 Corollary If L/K is a Galois extension of degree d, Br(L/K) is cyclic of order d.

Proof. Since the Br(L/K) ' H2(G, L×) is annihilated by d it must contained in the cyclic subgroup (1/d)Z/Z ⊆ Q/Z ' Br(K) of order d. Since this subgroup is the image of wL/K , invK maps Br(L/K) onto this subgroup. The fundamental class of the extension L/K is the element of Br(L/K) with invariant −1/d, where d = [L : K]. We identify this class with the elements of Ext(G, L×) and H2(G, L×) corresponding to it under the isomorphism 4.2.4.2, and in any case it will be denoted by uL/K . Our definition here differs from that of most of the classical texts (e.g. [1], [12]) but coincides with that of Tate [14] and Deligne [5]. In particular the reciprocity map to be defined later has the opposite sign to that found in the older texts. Our next is to give an explicit realization of the extension of G by L× rep- × resented by uL/K , or equivalently an explicit L -valued 2-cocycle representing uL/K . The division algebra corresponding to uL/K is the endomorphism ring of an isopentic F -isocrystal of rank d and slope 1/d, so we first need to construction some endomorphisms of VK (1/d). In fact the same method works for any VK (λ).

5.2.2.4 Lemma The sequence

× × σ−1 × wL/K −1 1 → L → LK −−−→ LK −−−−→ d Z → 0 is exact.

Proof. The only nonobvious point is that the image of σ − 1 is kernel of wL/K , and it is clearly contained in the kernel. We ﬁrst observe that the subgroup of f P Z consisting of (ni) such that i ni = 0 is spanned by elements with a 1 in position i, −1 in position i + 1 (addition modulo f) and zeros elsewhere. From × this it follows that any (xi) ∈ LK in the kernel of wL/K is congruent modulo f the image of σ − 1 to a (xi) such that vL(xi) = 0 for all x. Since σ − 1 is a retraction of (Lnr )× onto its subgroup of elements of valuation zero, for each i f f × there is a ci such that bi = σ (ci)/ci. Then b = σ (c)/c in LK , and since σf − 1 = (σ − 1)(σf−1 + ··· + σ + 1) we conclude that b is in the image of σ − 1.

s−1 Suppose now L/K is Galois with group G. For any s ∈ G, wL/K ( α) = 0 × and the lemma shows that there is a βs ∈ LK such that

σ−1 s−1 βs = α (5.2.2.3) well deﬁned up to a factor in L×. From 5.2.2.3 we see that

s us : LK → LK us(x) = βs · x (5.2.2.4) 158 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

commutes with Fα: σ s Fα(us(x)) = α · (βs · x) σ σs s sσ = α · βs · x = α · βs · x s σ = βs · (α · x) = us(Fα(x)). and is thus an automorphism of LK (α). A quick calculation shows that the us satisfy the relations s usut = as,tust, us` = ùs (5.2.2.5) for all ` ∈ L and s, t ∈ G, where s −1 as,t = βtβst βs. (5.2.2.6) The 5.2.2.5 are the definiing relations for the crossed product algebra defined by the cocycle 5.2.2.6. The correspdonding extension class can be deduced from 5.2.2.6, but it can also be described directly.

5.2.2.5 Theorem The endomorphism algebra DL/K (α) is isomorphic to the crossed product algebra associated to the 2-cocycle 5.2.2.6 associated to α. The extension corresponding to DL/K (α) is isomorphic to

× i p 1 → L −→ W (α) −→ GL/K → 1 × where W (α) the set of pairs (β, s) ∈ LK × G satisfying σ−1β = s−1α (5.2.2.7) with the composition law (β, s)(γ, t) = (β · sγ, st) (5.2.2.8) and i, p are the evident inclusion and projection.

Proof. Only the second part needs to be explained. In fact if i : K → DL/K (α) is the canonical injection, the extension group corresponding to DL/K (α) is the set of u ∈ DL/K (α) satisfying i ◦ s = ad(u) ◦ i. It is easily checked that these × have the form u = βs with β ∈ LK satisfying 5.2.2.7.

5.2.2.6 Corollary For any ﬁnite Galois extension L/K of degree d and any × α ∈ LK such that wL/K (α) = 1/d, the 2-cocycle 5.2.2.6 represents the fundamental class of L/K.

When wL/K (α) = 1/d, the extension W (α) representing the fundamental class is called the Weil group of L/K and denoted by WL/K . Up to isomorphism of extensions it is independent of the choice of α; in fact since H1(G, L×) = 0 it is well-defined up to inner automorphisms by elements of L× (c.f. the discussion in section 3.1.4). We note finally that just as the Galois group Gal(L/K) is (by definition!) an automorphism group, the same is true for W (α): by construction it consists of automorphisms u of LK (α) satisfying i ◦ s = ad(u) ◦ i for some s ∈ G. When K = Qp, this observation is due to Morava [10]. 5.2. THE FUNDAMENTAL CLASS 159

5.2.2.7 The unramiﬁed case again. Suppose L/K is unramiﬁed, so that f = d and we identify Knr ' Lnr . The Galois group G = Gal(L/K) is generated by the “arithmetic Frobenius” σarith = 1 ⊗ (σ|L) and we identify G ' Z/dZ by means of the generator σarith ; in the formulas to follow we also identify elements of Z/dZ with integers in the range [0, d). Choose a ∈ K× and let α = (a−1, 1,..., 1). Then 5.2.1.8 yields

σi −1 −1 arith α = (a, 1,..., 1, a , 1,..., 1) ↑ d−i

i × σ−1 σarith −1 and a βi ∈ LK satisfying βi = α is −1 −1 −1 βi = (a , a , . . . , a , 1,..., 1). (5.2.2.9) ↑ d−i−1 From this it follows easily that

( −1 i −1 a i + j < d βi(σ (βj))β = . arith i+j 1 i + j ≥ d The coboundary of the constant 1-cochain with value a is the constant 2-cocycle with value a. Adding this to the previous cocycle yields the 2-cocycle ( 1 i + j < d ai,j = a i + j ≥ d that we found earler. A class in Br(L/K) gives rise, via the isomorphisms

2 × 2 × Br(L/K) ' H (G, L ) ' ExtG(Z,L ) (5.2.2.10) to a Yoneda 2-extension of Z by L×. We can identify the extension corresponding to the fundamental class:

5.2.2.8 Theorem Suppose [L : K] = d and write w = wL/K . The class of uL/K ∈ Br(L/K) is the the class of the extension

× × σ−1 × dw 0 → L → LK −−−→ LK −−→ Z → 0. (5.2.2.11) 2 × in ExtG(Z,L ). Proof. We know from lemma 5.2.2.4 that the sequence is exact. Recall from 2 × section 4.1.2 how we associate a class in ExtG(Z,L ) to the 2-extension 5.2.2.11: if P· → Z is a resolution of Z by projective Z[G]-modules, there is a morphism of complexes

d3 d2 d1 d0 / P3 / P2 / P1 / P0 / Z / 0 (5.2.2.12)

f3 f2 f1 f0 × × σ−1 × dw / 1 / L / LK / LK / Z / 0 160 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

× unique up to homotopy. The map f2 lies in the kernel of Hom(P2,L ) → × 2 × Hom(P3,L ), and its image in H (G, L ) is the class of 5.2.2.11. For P· we take the bar complex and recall that

d0([ ]) = 1, d1([s]) = s[] − [], d2([s|t]) = s[t] − [st] + [s]. If we set f0([ ]) = α, f1([s]) = βs, f2([s|t]) = as,t the commutativity of 5.2.2.12 shows that σ−1 s−1 s −1 dw(α) = 1, βs = α, as,t = βtβst βs.

The ﬁrst equality shows that DL/K (α) represents the fundamental class uL/K , 2 × and the others show that (as,t) represents DL/K (α) in H (G, L ). Since the class of 5.2.2.11 is (as,t) we are done.

5.3 The Reciprocity Isomorphism

5.3.1 The inverse reciprocity map. The next proposition and its corollary are essentially exercise 2 of [12, Ch. XIII §5]:

× 5.3.1.1 Proposition The G-module LK is cohomologically trivial. × Proof. We first observe that the G-module LK is induced from the I-module (Lnr )×, where I ⊂ G is the inertia subgroup. Replacing L/K and G by Lnr /Knr and I respectively, we are reduced to showing that if L/K is totally ramified and the residue field of K is algebraically closed, L× is a cohomologically trivial G-module. This is theorem 3.4.2.4.

5.3.1.2 Corollary For all r ∈ Z, the map r uL/K ∪ r+2 × Hˆ (G, Z) −−−−→ Hˆ (G, L ) (5.3.1.1) is an isomorphism. Proof. Split the extension 5.2.2.11 into the short exact sequences × × 1 → L → LK → U → 1 × 1 → U → LK → Z → 0 and denote by 1 × 1 u1 ∈ ExtG(U, L ), u2 ∈ Ext (Z,U) 2 × the corresponding extension classes; then uL/K ∈ ExtG(Z,L ) is the product u1 · u2 for the composition 1 × 1 1 × ExtG(U, L ) × ExtG(Z,U) → ExtG(Z,L ).

On the other hand the cup products with u1 and u2 are the connecting homomorphisms for the corresponding exact sequences, which by the proposition are isomorphisms on Tate cohomology . 5.3. THE RECIPROCITY ISOMORPHISM 161

5.3.2 The norm residue symbol. We denote by

ab × × ηL/K : G → K /NL/K (L ) (5.3.2.1) the isomorphism 5.3.1.1 for r = −2; we will also use this to denote the induced homomorphism × × ηL/K : G → K /NL/K (L ). The reciprocity map

−1 × × ab θL/K = ηL/K : K /NL/K (L ) → G (5.3.2.2) is the inverse of ηL/K . We will also write θL/K for the composite

× × × ab K → K /NL/K L → G .

The traditional notation for θL/K notation, which is

(a, L/K) = θL/K (a) (5.3.2.3) × × × where a is indiﬀerently an element of K or K /NL/K L . The left hand side of 5.3.2.3 is also known as the norm residue symbol or the Artin symbol.

5.3.2.1 Theorem Suppose L/K is a Galois extension of local ﬁelds with group × G and set d = [L : K]. Let α be an element of LK such that wL/K (α) = 1/d, × and let s ∈ G. If β ∈ LK satisﬁes σ−1β = s−1α then × ηL/K (s) = NL/K (β) mod NL/K L . (5.3.2.4)

Proof. It suffices to recall the formulas for the isomorphism H−2(G, Z) ' Gab and the procedure for computing the cup product with uL/K . The first identifies the image s¯ of s ∈ G in Gab with the class of ˇ [s] ⊗G 1 ∈ B1 ⊗G Z = C1(G, Z) = Z1(G, Z) −2 ∗ in H1(G, Z) = H (G, Z). As in 3.3.2.8 we identify B−n−1 ' Bn; then the homomorphism f : B−2 → Z representing s¯ is ( 1 u = s f : B1 → Z f([u]) = . 0 u 6= s ¯ This vanishes on dB0 and we denote by f : B1/dB0 → Z the induced map. Then f¯ extends to a morphism of complexes

d−1 d−2 B1 / B0 / B0 / B1 / B1/dB0 / 0

f2 f1 f0 f¯ dw × × σ−1 × L/K 0 / L / LK / LK / Z / 0 162 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

and f0 can be any map making the rightmost square commutative. We choose ( α u = s au = f0([u]) = 1 u 6= s

× where α is any element of LK such that dwL/K (α) = 1. If we set β = f1([ ]) and c = f2([ ]) and use the formulas X X d−1([ ]) = s[], d−2([ ]) = (s − 1)[s] s∈G s∈G from 3.3.2.9 and 3.3.2.10 we ﬁnd

σ−1 s−1 β = α, c = NG(β) = NL/K (β).

× × Finally uL/K ∪ s¯ is the image in K /NL/K L of c = f2([ ]), which completes the proof.

5.3.2.2 Corollary If L/K is unramiﬁed and π is a uniformizer of K,

−1 × ηL/K (σarith ) = π mod NL/K L . (5.3.2.5)

Proof. To compute ηL/K (σarith ) we can use

β = (π−1, 1,..., 1)

−1 as in 5.2.2.9, and then NL/K (β) = π . In more traditional notation, this is

−1 (π, L/K) = σarith (5.3.2.6) when L/K is unramiﬁed. As we remarked before, the traditional normalization of the norm residue symbol would have σarith on the right hand side. nr −1 When L/K is totally ramiﬁed we may identify LK = L and take α = π for any uniformizer π of Lnr ; in this case 5.3.2.4 is equivalent to the formula proven by Dwork [7]. The general formula can be deduced from a series of exercises in [12, Ch. XIII §5].

5.3.3 Successive extensions. In order to find the behavior of the norm residue symbol with respect to field automorphisms and successive extensions we need to extend the formalism developed in section 5.2.1. In the next few sections we fix a finite Galois extension E/K with group GE/K and a subfield L of G containing K. The Galois group of E/L will be denoted by GE/L, and if L/K is Galois its group is GL/K . nr nr The transitivity isomorphism K ⊗K E ' (K ⊗K L)⊗L E may be written

EK ' LK ⊗L E. (5.3.3.1) 5.3. THE RECIPROCITY ISOMORPHISM 163

This shows that EK has a canonical structure of a free LK -module, and we denote by K × × NE/L : EK → LK (5.3.3.2) the norm map. When E/L/K is L/K/K this is the norm map 5.2.1.9 introduced K earlier. The norm NE/L is equivariant for the action of σ on EK and LK . If E/K is Galois with group GE/K , it is also equivariant for the action of GE/K in the sense that N K × E/L × EK / LK (5.3.3.3)

s s E× s(L)× K K / K NE/s(L) commutes for any s ∈ GE/K . Finally, for a 3-fold extension F/E/L/K the norm is transitive: K K K NE/L ◦ NF/E = NF/L. (5.3.3.4) Applying this in the case E/L/K/K and invoking 5.2.1.11, we ﬁnd that

wL/K (NE/L(β)) = [E : L]wE/K (β) (5.3.3.5) for β ∈ EK . The decomposition 5.2.1.3 extends in an obvious way to a decomposition of nr EK as an EL-module. As before we ﬁx a completion L of a maximal unram- iﬁed extension of L and a morphism Knr → Lnr making 5.2.1.2 commutative. Tensoring the isomorphism of L with E yields a direct sum decomposition

M i ∗ nr M i ∗ EK ' LK ⊗L E ' (σ ) L ⊗L E ' (σ ) EL (5.3.3.6) 0≤i

nr of K ⊗K E-bimodules. This direct sum decomposition is, like 5.2.1.7 canonically determined by a choice of diagram 5.2.1.2. The isomorphism 5.3.3.6 is equivariant for the action of GE/L on both sides. K L For any 3-fold extension F/E/L/K of local ﬁelds the norms NF/E and NF/E are related as follows: if

M i ∗ M i ∗ FK ' (σ ) FL EK ' (σ ) EL 0≤i

M i ∗ γ = (γ0, γ1, . . . γf−1) ∈ FK ' (σ ) FL 0≤i

for the above decomposition of EK . This can be seen by ﬁxing a basis of F as an E-vector space and using the fact that determinants relative to a direct sum of rings respect the direct sum decomposition. Recall that for the decomposition 5.2.1.3 of LK the action of σ is given by nr 5.2.1.6 where σL is the Frobenius of L with respect to its own residue ﬁeld. nr It follows that the action of σ as σ ⊗ 1 on EK = K ⊗K E is then given by

(β0, β1, . . . , βf−1) 7→ (σL(βf−1), β0, β1, . . . , βf−2) (5.3.3.8) for the decomposition 5.3.3.6, where now σL denotes the extension of σL acting nr on L to σL ⊗ 1 acting on EL. We now deﬁne homomorphisms

E E × × ιL/K , δL/K : EL → EK by

E ιL/K (α) = (α, 1,..., 1) (5.3.3.9) E δL/K (β) = (β, β, . . . , β) with respect to the decomposition 5.3.3.6. Since the decomposition 5.3.3.6 is E E equivariant for the action GE/L, so are ιL/K and δL/K . Corresponding to the × formula 5.3.3.5 we have, for α ∈ EL , 1 w (ιE (α) = w (α). (5.3.3.10) E/K L/K [L : K] L/K For any 3-fold extension F/E/L/K the maps ι and δ are compatible with the norms. In fact 5.3.3.7 implies that the diagrams

δF ιF × L/K × × L/K × FL / FK FL / FK (5.3.3.11)

L K L K NF/E NF/E NF/E NF/E E× E× E× E× L E / K L E / K δL/K ιL/K are commutative. We need one more fact about these maps. The composite

δL N K × L/K × L/K × LL −−−→ LK −−−−→ KK is a homomorphism (Lnr )× → (Knr )×. It is not the usual norm map, however:

5.3.3.1 Lemma For ` ∈ L× ⊂ (Lnr )×,

K L NL/K (δL/K (`)) = NL/K (`) (5.3.3.12) where the norm on the right is the ﬁeld norm for L/K. 5.3. THE RECIPROCITY ISOMORPHISM 165

K Proof. Since NL/K is the map 5.2.1.9, the equality 5.2.1.10 shows that

K L Y i NL/K (δL/K (`)) = σ (NLnr /Knr (`)). 0≤i

As before let L0 be the maximal unramiﬁed extension of K in L, and identify nr nr nr nr nr it with a subﬁeld of K . Since L ' L ⊗L0 K , NL /K (`) = NL/L0 (`) and the product on the right hand side is

Y i σ (NL/L0 (`)) = NL0/K (NL/L0 (`)) = NL/K (`) 0≤i

5.3.4 Functorial properties. We can now express the behavior of the norm residue symbol with respect to ﬁeld automorphisms and successive extensions, starting with the latter. In the above situation we suppose that L/K is Galois, and denote by K πE/L : GE/K → GL/K × the canonical homomorphism. To calculate ηE/K we choose α ∈ EK such that × σ−1 s−1 wE/K (α) = 1/[E : K]: if s ∈ GE/K we pick β ∈ EK such that βs = α; K then ηE/K (s) = NL/K (βs) (note that NL/K = NL/K here). On the other hand 5.3.3.5 says that

[E : L] 1 w (N K (α)) = = L/K E/L [E : K] [L : K] and we may use NE/L(α) to calculate ηL/K . By equivariance

σ−1 s−1 NE/L(β) = NE/L(α) (5.3.4.1) so from 5.3.3.4 and 5.3.2.4 we get

K × ηE/K (s) = ηL/K (πE/L(s)) mod NL/K L (5.3.4.2) or equivalently that

K πE/L GE/K / GL/K (5.3.4.3)

ηE/K ηL/K

× × × × K /NE/K E / K /NL/K L where the bottom arrow is the natural map. In the more traditional notation 5.3.2.3 this says that K πE/L(a, E/K) = (a, L/K) (5.3.4.4) 166 CHAPTER 5. THE RECIPROCITY ISOMORPHISM or in other words the norm residue symbol is compatible with passage to a larger Galois extension of K. We next allow L to be any intermediate extension and take s ∈ GE/L. To × −1 × compute ηE/L(s) we choose α ∈ EL such that wE/L(α) = [E : L] and β ∈ EL σL−1 s−1 × × such that β = α; then ηE/L(s) is the class of NE/L(β) in L /NE/LE L (recall here that NE/L here is NE/L). On the other hand by 5.3.3.10 we have E E wE/K (ιL/K (α)) = 1/[E : K], so we can use ιL/K (α) to compute ηE/K (s). The formula 5.2.1.6 for the action of σ on LK shows that

σ−1 K σL−1 s−1 s−1 K δE/L(β) = ( β, 1,..., 1) = ( α, 1,..., 1) = ιE/L(α) and thus K E × ηE/K (s) = NE/K (δL/K (β)) mod NE/K E . To evaluate the right hand side we consider the diagram

δE × L/K × E E K L / K NE/K

L K NE/L NE/L L× L× K× L L / K K / K δL/K NL/K in which the square is the case E/L/L/K of the commutative square on the left of 5.3.3.11, and the right hand triangle commutes by the transitivity of norms. From this we see that

K K K L L K L NE/K (δE/L(β)) = NL/K (δL/K (NE/L(β))) = NL/K (δL/K (ηE/L(s))).

× Since NE/L(βL) ∈ L , lemma 5.3.3.1 shows that

K L NL/K (δL/K (ηE/L(s))) = NL/K (ηE/L(s)) and combining the previous equalities yields

× ηE/K (s) = NL/K (ηE/L(s)) mod NE/K E . (5.3.4.5)

In other words i GE/L / GE/K (5.3.4.6)

ηE/L ηE/K

× × × × L /NE/LE / K /NE/K E NL/K commutes; in terms of the norm residue symbol this is

(a, E/L) = (NL/K (a), E/K) (5.3.4.7) for a ∈ L×. 5.4. WEIL GROUPS 167

σ−1 s−1 Again with s, t ∈ GE/K and βs = α, the equality

σ−1 t t σ−1 ts−t tst−1−1 t ( βs) = ( βs) = α = ( α) implies that −1 t ηE/t(L)(tst )) = ηE/L(s) (5.3.4.8) modulo appropriate subgroups, i.e.

ad(t) GE/L / GE/t(L) (5.3.4.9)

ηL/K ηLt/K

× × × × L /NE/LL / K /NE/t(L)(t(L))

−1 −1 commutes, where ad(t): GE/L → GE/t(L) = tGE/Lt is s 7→ tst . The corresponding formula for the norm residue symbol is

s(a, E/L)s−1 = (s(a), E/s(L)) (5.3.4.10) for s ∈ GE/K and a ∈ L. ab ab The last compatibility asserts that the transfer Ver : GE/K → GL/K corre- × × × × sponds via the norm residue symbol to the map K /NE/K E → L /NE/LE induced by the inclusion. From our point of view this is best understood in terms of the ideas of the next section (proposition 5.4.1.4).

5.4 Weil Groups

The Weil groups have a formalism relative to successive extensions similar to that of the Galois groups. This formalism is related to and in fact re-expresses the functorial properties of the norm residue symbol that were worked out in the last section.

5.4.1 The formalism of Weil groups. We fix a Galois extension E/K with group GE/K and let L be an extension of K in E (not necessarily Galois over K). As before GE/L is the Galois group of E/L, and similarly for GL/K when L/K is Galois. Our aim in this section is to define morphisms i : WE/L → WE/K and, when L/K is Galois, πE/L : WE/K → WL/K . Since WL/K by definition is only defined up to inner automorphisms it will be necessary to “rigidify” it by a particular choice of “realization” W (α), which as before is × the set of pairs (β, s) ∈ LK × GL/K satisfying 5.2.2.7 with the composition law 5.2.2.8. To indicate the fields involved we write WL/K (α) for W (α), so a homomorphism such as πE/L : WE/K → WL/K should be understood as a 0 homomorphism WE/K (α ) → WL/K (α) for particular choices (usually implicit) of α, α0 appropriate to L/K and E/K. 168 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

In fact everything we need is contained in the formulas 5.3.4.1 and 5.3.4.5. × × The ﬁrst of them shows that NE/L : EK → EL extends to a homomorphism

π : W (α) → W (N (α)) E/L E/K L/K E/L (5.4.1.1) πE/L(β, s) = (NE/L(β), πL/K (s)) making commutative the diagram

× 1 / E / WE/K / GE/K / 1 . (5.4.1.2)

NE/L πE/L πE/L

× 1 / L / WL/K / GL/K / 1

The transitivity of the norm shows that the maps π just constructed are transitive: K K K πE/L ◦ πF/E = πF/L (5.4.1.3) for a 3-fold extension F/E/L/K; here of course the groups WF/K , WE/K and WL/K are realized as WF/K (α), WE/K (NF/L(α) and WL/K (NF/L(α) for appropriate α; from now on we will not be explicit about this. × In the special case when E/L/K is L/K/K, we have WK/K = K , and the K deﬁnition of πL/K shows that the diagram

WL/K / GL/K (5.4.1.4)

K η πL/K L/K

× × K / K/NL/K L commutes, where the horizontal maps are the natural projections. In fact if × σ−1 s−1 α ∈ LK is used to deﬁne WL/K , (β, s) ∈ WL/K implies that β = α, so that

× × (πL/K (β, s) mod NL/K L ) = (NL/K (β) mod NL/K L ) = ηL/K (s).

That 5.4.1.4 is commutative could be expressed by saying that πL/K : WL/K → K× is a lifting of the inverse reciprocity map. It could also be rephrased as saying that the diagram

πab ab L/K × WL/K / K (5.4.1.5)

θL/K # ab GL/K is commutative. 5.4. WEIL GROUPS 169

× × Now 5.3.4.5 shows that δE/L : LK → EK extends to a homomorphism

K iE/L : WL/K (α) → WE/K (ιE/L(α)) (5.4.1.6) K iE/L(β, s) = (δE/K (β), s) making commutative a diagram

× 1 / L / WL/K / GL/K / 1 (5.4.1.7)

K iE/L

× 1 / E / WE/K / GE/K / 1 where the extreme vertical arrows are the canonical inclusions. For a 3-fold extension F/E/L/K we have

K K K δF/E ◦ δE/L = δF/L (5.4.1.8) which shows that the i are transitive:

K K K iF/E ◦ iE/L = iF/L. (5.4.1.9)

Finally the right hand diagram of 5.3.3.11 implies that for any 3-fold extension F/E/L/K the diagram

K πF/E WF/L / WE/L (5.4.1.10)

K K iL/K iL/K W W F/K K / E/K πF/E is commutative. This can be expressed by saying that if we use the maps iE/L to identify WE/L with a subgroup of WE/K , then these identiﬁcations are compatible with the canonical projection maps π (assuming as always that compatible choices of α are made in each place). K It is evident from the construction that iE/L : WE/L → WE/K is injective. It K is also true that πE/L : WE/K → WE/L is surjective, but this is not so obvious.

× × 5.4.1.1 Proposition The connecting homomorphism ∂ : GE/L → L /NE arising from the diagram 5.4.1.2 is the inverse norm residue homomorphism ηE/L.

Proof. For s ∈ GE/L, ∂(s) is computed as follows: lift s to an element (β, s) ∈ × WE/K ; then πE/L(β, s) ∈ WL/K lies in L ⊂ WL/K and ∂(s) is the image × × of πE/L(β, s) in L /NE/LE . The commutative diagram 5.4.1.10 applied to 170 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

E/L/L/K is πE/L WE/L / WL/L

iL/K iL/K WE/K / WL/K πE/L which shows that ∂(s) can also be computed by lifting s to (β, s) ∈ WE/L, × × × and then ∂(s) is the class of πE/L(β, s) ∈ WL/L = L in L /NE/LE . Now × nr × × πE/L(β, s) = NE/L(β) where NE/L is the norm EL → (L ) , and if α ∈ EL σ−1 s−1 is used to deﬁne WE/L, β satisﬁes β = α, so ∂(s) = ηE/L(s) by Dwork’s formula.

5.4.1.2 Corollary For any successive extension E/L/K of local ﬁelds with K E/K and L/K Galois, the homomorphism πE/L : WE/K → WL/K is surjective.

Proof. Since ∂ = ηE/L is surjective, this follows from the snake lemma.

5.4.1.3 Lemma For any Galois extension of local ﬁelds the transfer Ver : ab × WL/K → L is the composite

πab ab L/K × × WL/K −−−→ K → L

× where we identify WK/K ' K , and the second map is the inclusion. Proof. Recall the definition of the transfor Gab → Hab for a subgroup H ⊆ G of finite index: choose a section θ : H \ G → G of the projection G → H \ G, and for s, t ∈ G define xs,t ∈ H by θ(Ht)s = xt,sθ(Hts); then Y Ver(s) = xt,s t∈H\G

ab × where the product is in H . In our case G = WL/K and we identify H = L with its abelianization. For t ∈ GL/K = G/H we take θ(t) = (βt, t) for some × appropriate βt ∈ LK ; then

t t −1 θ(t)(β, s) = (βt, t)(β, s) = (βt · β, ts) = (βt ββts , 1)θ(ts)

t −1 × where the last equality is justiﬁed since βt ββts ∈ L . For this θ then we have s xt,s = ( β, 1), and then

Y t −1 Ver(β, s) = (βt ββts , 1) = (NL/K (β), 1)

t∈GL/K which proves the assertion. We can now prove the last compatibility for the norm residue symbol. 5.4. WEIL GROUPS 171

5.4.1.4 Proposition For any 2-fold extension E/L/K of local ﬁelds with E/K Galois, K× / L× (5.4.1.11)

θE/K θE/L ab Ver ab GE/K / GE/L is commutative, where the upper horizontal map is the inclusion.

Proof. In view of the diagram 5.4.1.5 applied to E/K and E/L, and since for × any Galois extension L/K the projection πE/K : WE/K → K is surjective, it suﬃces to show that the diagrams

ab Ver ab ab Ver ab WE/K / WE/L WE/K / WE/L (5.4.1.12)

ab ab πE/K πE/L

× × Gab / Gab K / L E/K Ver E/L incl are commutative. The ﬁrst just expresses the functoriality of the transfer. The second can be embedded in the diagram

ab Ver ab WE/K / WE/L

πab πab E/K E/L Ver K× / L×

j # i 0 E×

ab ab in which i and j are the inclusions. By lemma 5.4.1.3 i ◦ πE/K and j ◦ πE/L are × × the transfer homomorphisms for the subgroups E ⊂ WE/K and E ⊂ WE/L respectively. Since the transfer is transitive for chains of subgroups, the outside square in this diagram is commutative, and since L× → E× is injective, the inside square is as well. In terms of the norm residue symbol, the commutativity of 5.4.1.11 says that

Ver(a, E/K) = (a, E/L) (5.4.1.13) for a ∈ K×. Proposition 5.4.1.4 is less elementary than the previous compatibili- ab ab ties since the homomorphism GE/K → GE/L is not induced by a homomorphism GE/K → GE/L. The real reason that Weil groups behave similarly to Galois groups relative to successive extensions is that the latter are in fact quotients of the former, as 172 CHAPTER 5. THE RECIPROCITY ISOMORPHISM was shown by Shafarevich [13]. Suppose that E/L/K are extensions of local ﬁelds with E/K and L/K Galois. We deﬁne a map

× Sh : GE/K → WL/K /NE/LE (5.4.1.14) as follows: for s ∈ GE/K choose x ∈ WE/K mapping to s under the natural K projection WE/K → GE/K . Then πE/L(x) ∈ WL/K is well-deﬁned up to a factor × × K in NE/LE ⊂ L ⊂ WL/K , and we deﬁne Sh(s) to be the class of πE/L(x) in × WL/K /NE/LE .

5.4.1.5 Theorem (Shafarevich) Suppose that E/L/K are extensions of local ﬁelds with E/K and L/K Galois. The diagram

1 / GE/L / GE/K / GL/K / 1 (5.4.1.15)

ηE/L Sh

× × × 1 / L /NE/LE / WL/K /NE/LE / GL/K / 1 is commutative. In particular if E/L is abelian, Sh is an isomorphism.

Proof. The commutativity of the right hand square is clear from the construction. As to the one on the left, suppose WE/L has been deﬁned by some × E α ∈ EL and that WE/K is deﬁned by ιL/K (α), as in our construction of i : WE/L → WE/K . Then if s ∈ GE/L, any lifting of it to WE/K lies in the E E × image of iL/K and we may suppose it has the form (δL/K (β), s), with β ∈ LK satisfying σ−1β = s−1α. Now the diagram 5.4.1.10 in the case when F/E/L/K is E/L/L/K is L πE/L WE/L / WL/L

E L iL/K iL/K W W E/K K / L/K πE/L which shows that Sh(s) can be identiﬁed with the image of

K πE/L(β, s) = (NE/L(β), 1) ∈ WL/K .

Since NE/L(β) = ηE/L(s) the left hand square is commutative. Finally if E/L is abelian ηE/L is an isomorphism, hence so is Sh.

K 5.4.2 The kernel of πE/L. Suppose E/L/K are successive Galois exten- K sions of local ﬁelds with L/K Galois. We have seen that πE/L : WE/K → WL/K is surjective. The last part of the picture to ﬁll in is the kernel. We need the following algebraic result of Artin and Tate [1, Ch. 13 Thm. 3]: 5.4. WEIL GROUPS 173

5.4.2.1 Proposition Suppose

0 → A → E → G → 0 is an extension of a ﬁnite group G by an abelian group A with cocycle η ∈ H2(G, A), and denote by N : A → AG norm for the action of G on A. Then

c IGA ⊆ E ∩ A ⊆ Ker(N) and the image of

−3 −1 ∪η : H (G, Z) → H (G, A) = Ker(N)/IGA

c is (E ∩ A)/IGA.

c Proof. It is clear that IGA and E ∩ A are contained in Ker(N). To show that c IGA ⊆ E it suﬃces to observe that if the image of e ∈ E is s then

eae−1a−1 = saa−1 = s−1a.

To prove the last statment it suﬃces to show that a character (i.e. homomorphism to Q/Z) of A vanishes on Ec ∩ A if and only if it vanishes on the inverse image in Ker(N) of the image of ∪η. We will establish that each of the following conditions are equivalent to f vanishing on Ec ∩ A:

• f extends to a character of E: in fact any character of E necessarily vanishes on Ec ∩ A; on the other hand if f vanishes on Ec ∩ A it deﬁnes a character of A/(Ec ∩ A) ' AEc/Ec, which then extends to a character of E since Q/Z is an injective Z-module.

• The pushout of the extension by f splits, or in other words f∗(η) = 0. In fact a splitting yields an extension of f to E; conversely an extension of f to a character of E yields a splitting by the universal property of pushouts.

ˆ −3 2 • (f∗η)(ξ) = 0 for all ξ ∈ H (G, Z): this follows since H (G, Q/Z) is the ˆ −3 linear dual of H (G, Z). Since (f∗η)(ξ) = f∗(η ∪ ξ), another equivalent condition is

ˆ 3 • f∗(η ∪ ξ) = 0 for all ξ ∈ H (G, Z). In other words, f∗ vanishes on the subgroup −3 −1 η ∪ Hˆ (G, Z) ⊆ Hˆ (G, A).

The last condition, ﬁnally is equivalent to that of the proposition.

5.4.2.2 Corollary For any Galois extension L/K of local ﬁelds,

× × c × Ker(NL/K : L → K ) = WL/K ∩ L . (5.4.2.1) 174 CHAPTER 5. THE RECIPROCITY ISOMORPHISM

Proof. In this case η = uL/K and the cup product with uL/K is an isomorphism.

5.4.2.3 Theorem Suppose E/L/K are Galois extensions of local ﬁelds with K E/K Galois. The homomorphism iE/L : WE/L → WE/K identiﬁes the kernel K c of πE/L with the commutator subgroup WE/L. In particular there is an isomorphism c ∼ WE/K /WE/L −→ WL/K . (5.4.2.2)

K Proof. Note ﬁrst that any of the possible choices for iE/L are conjugate by × c an element of L ⊂ WE/L, so the image of WE/L in WE/K is ﬁxed. The commutative diagram 5.4.1.2 shows that the kernel of WE/K → WL/K maps to the kernel of GE/K → GL/K and is therefore contained in WE/L. The diagram 5.4.1.10 applied to E/L/L/K is

L πE/L WE/L / WL/L

E L iL/K iL/K W W E/K K / L/K πE/L

K K and since the vertical arrows are injective, iE/L identiﬁes the kernel of πE/L × with the kernel of WE/L → WL/L → WL/K . Since WL/L = L is abelian, c WE/L is contained in the kernel. To prove the converse inclusion we apply the snake lemma to the diagram 5.4.1.2. The result is yields an exact sequence

K ηE/L × × 0 → Ker(NE/L) → Ker(πE/K ) → GL/K −−−→ L /NE/LE → 0

× × where NE/L is the field norm E → L , and we have used corollary 5.4.1.2 and c proposition 5.4.1.1. We know that the kernel of ηE/L is GE/L. Suppose now x ∈ K Ker(πE/L); by the last remark its image in GE/L is in the commutator subgroup, c so x is the product of an element of WE/L and an element of Ker(NE/L). But c corollary 5.4.2.2 shows that elements of Ker(NE/L) are in WE/L as well, and we are done. Note that the isomorphism 5.4.2.2 depends on the specific identifications of × the groups WE/K and WL/K , i.e. the choice of α ∈ EK . As remarked in the c proof however, the subgroup WE/L ⊂ WE/K is independent of such choices. Chapter 6

The Existence Theorem

The first result of (global) class field theory was the Kronecker-Weber theorem, which states that any abelian extension of Q is contained in a cyclotomic ex- ¯ tension, i.e. some field Q(µn) there µn is the group of nth roots of unity in Q. In other words the abelian extensions are generated by “torsion points” of the multiplicative group, and Kronecker was able to show similarly that all abelian extensions of an imaginary quadratic extension of Q could be generated from torsion points on suitable elliptic curves. The generalization to other number fields remains problematic. In the local case the situation is more satisfactory. Any abelian extension of Qp is cyclotomic, i.e. contained in an some extension Qp(µn) where now µn is the ¯ group of nth roots of unity in Qp. For general local fields in characteristic zero, Lubin and Tate were constructed formal groups whose torsion points generate totally ramified abelian extensions of the given local field. The existence theorem will be deduced from this, and the Hasse-Arf theorem for local fields is an easy consequence.

6.1 Formal Groups

6.1.1 Formal Group Laws. Let R be a commutative ring (with identity, as always). A one-dimensional formal group law over R is a power series in two variables F (X,Y ) ∈ R[[X,Y ]] such that

F (X,Y ) ≡ X + Y mod (X,Y )2, (6.1.1.1) F (X, 0) = F (0,X) = X, (6.1.1.2) F (X,Y ) = F (Y,X), (6.1.1.3) F (F (X,Y ),Z) = F (X,F (Y, x)). (6.1.1.4)

One can show that 6.1.1.2 follows from 6.1.1.1 and 6.1.1.4. Furthermore 6.1.1.3 is a consequence of the remaining conditions if R has no Z-torsion, but we will not need this fact. One can also deduce from 6.1.1.1 that there is a unique

175 176 CHAPTER 6. THE EXISTENCE THEOREM power series i(X) such that

F (X) ≡ −X mod (X2),F (i(X),X) = F (X, i(X) = 0 (6.1.1.5) which fact we wiil leave as an exercise. Formal groups in several variables are deﬁned similarly. To cut down on the density of notation one sometimes uses the notation X +F Y = F (X,Y ). (6.1.1.6) for the formal group law F (X,Y ). If F and G are one-dimensional formal group laws a morphism F → G is a power series f(X) ∈ R[X] such that

f(X) +G f(Y ) = f(X +F Y ). (6.1.1.7) and as usual we say that f is an strict isomorphism if there is a power series g(X) such that f(g(X)) = g(f(X) = X. Then g is a morphism G → F , as one easily sees from 6.1.1.7. An isomorphism f is a strict isomorphism if f(X) ≡ X mod (X2. More generally, if A is a commutative R-algebra, then a morphism (resp. isomorphism, strict isomorphism) over A is a power series f ∈ A[[X]] satisfying 6.1.1.7 (resp. and is an isomorphism, resp. a strict isomorphism in the previous sense). As usual we denote by HomA(F,G) the set of morphisms F → G defined over A, and if A = R we just write Hom(F,G). Similarly EndA(F ) is the ring of endomorphisms of F defined over A, and if A = R we drop the subscript. We will see many examples later but for now just mention two: the formal ˆ additive group Ga for which ˆ Ga(X,Y ) = X + Y (6.1.1.8) ˆ and the formal multiplicative group Gm defined by ˆ Gm(X,Y ) = X + Y + XY = (1 + X)(1 + Y ) − 1. (6.1.1.9) The “power operations” for a formal group F are defined in the usual way:

[0](X) = 0, [n + 1](X) = [n](X) +F X forn ≥ 1 (6.1.1.10)

(we do not usually specify F in the notation). For example, in the additive ˆ group Ga [n](X) = nX (6.1.1.11) ˆ and in the multiplicative group Gm [n](X) = (1 + X)n − 1. (6.1.1.12)

If R is a topological ring it makes sense to speak of the convergence of F (X,Y ) for appropriate values of X and Y , in which case F defines a group law (assuming it is “closed” in an obvious sense). The main case of interest to us is 6.1. FORMAL GROUPS 177 when R has the I-adic topology for some ideal I ⊂ R. In this case it is clear from (6.1.1.1)-(6.1.1.4) that F induces a commutative group law on I which we ˆ denote by F (I). For example Ga(I) is isomorphic to I considered as an additive ˆ × subgroup of R, while Gm(I) is isomorphic to the subgroup 1 + I ⊆ R . We should emphasize though that the notion of a formal group law is a completely algebraic one, and that it makes sense, for example to speak of formal group law over Z. If R is a field of characteristic p > 0 one can show that either [p](X) is zero h or its first nonzero term is aXp for some positive integer h. In the first case we say the formal group law has infinite height, and in the second it has height h. For example the formal additive group over a field of characteristic p > 0 has infinite height, while the formal multiplicative group has height one. Later on we will construct formal groups of any given height. If R is the integer ring of a mixed characteristic valued field and F is a formal group law over R, the height of F is the height of its reduction modulo the maximal ideal of R.

6.1.2 Logarithms. Suppose R is an integral domain with fraction ﬁeld K and F (X,Y ) is a formal group law over R.A logarithm for F is a power series `(X) ∈ K[[X]] satisfying `(X) ≡ X mod (X)2, `(F (X,Y )) = `(X) + `(Y ). (6.1.2.1) A formal group law does not necessarily have a logarithm, and if it does it is not necessarily unique. The Lubin-Tate formal groups to be considered later do in fact have logarithms, and in some sense the logarithm is the more fundamental object than the formal group itself. Note that the ﬁrst condition in 6.1.2.1 implies that ` has a functional inverse, and thus the formal group law F is uniquely determined by `: F (X,Y ) = `−1(`(X) + `(Y )). (6.1.2.2) Suppose F and F 0 are formal group laws over R with logarithms `, `0.A power series h ∈ R[[X]] is a morphism F → F 0 if there is an a ∈ R such that `0 ◦ h = a` (6.1.2.3) as one sees immediately from 6.1.2.2. When 6.1.2.3 holds we have h(X) ≡ aX mod (X2), so if a ∈ R×, h is an isomorphism. The problems with existence and uniqueness only arise in positive characteristic.

6.1.2.1 Proposition Suppose R is an integral domain whose fraction field has characteristic 0. Any formal group over R has a unique logarithm. Proof. Suppose first that F is a formal group law over R and ` is a logarithm for F . If we differentiate the equality in 6.1.2.1 with respect to Y we find

0 0 ` (F (X,Y ))F2(X,Y ) = ` (Y ) 178 CHAPTER 6. THE EXISTENCE THEOREM

where F2(X,Y ) is the derivative F with respect to its second argument. Setting Y = 0 and using 6.1.1.2 yields

0 0 ` (X)F2(X, 0) = ` (0) = 1. (6.1.2.4)

The property 6.1.1.1 shows that F2(X, 0) ≡ 1 mod (X), so F2(X, 0) is an invertible power series. Thus `0 is determined by F , and since ` has no constant term it is itself determined by F . The equation 6.1.2.4 shows that in fact we should deﬁne `(X) by

Z X dX `(X) = (6.1.2.5) 0 F2(X, 0) where the right hand side is to be understood as the (unique) integral of the −1 power series F2(X, 0) . To show that the ` so defined satisfies 6.1.2.1 we 2 −1 first note that `(X) ≡ X mod (X ) since F2(X, 0) ≡ 1 mod (X). Second, differentiating the associativity law 6.1.1.4 with respect to Z yields

F2(F (X,Y ),Z) = F2(X,F (Y,Z))F2(Y,Z) and setting Z = 0 and using 6.1.1.2 results in

F2(F (X,Y ), 0) = F2(X,Y )F2(Y, 0).

If we write this as 1 1 ∂ = (F (X,Y )) F2(Y, 0) F2(F (X,Y ), 0) ∂Y and integrate with respect to Y we get

`(Y ) + g(X) = `(F (X,Y )) for some g ∈ K[[X]]. Switching X and Y and using the commutativity of the group law, we ﬁnd that g(X) = `(X), which proves 6.1.2.4.

6.1.2.2 Corollary If R is a Q-algebra then any formal group over R is iso- ˆ morphic to Ga. Proof. In this case the logarithm lies in R[[X]]. Equation 6.1.2.4 shows what can go wrong in positive characteristic: if F = ˆ Gm is the formal multiplicative group 6.1.2.4 says that a logarithm ` must satisfy 1 X `0(X) = = (−1)nXn 1 + X n>0 but in positive characteristic the series cannot be integrated. As for the question P pn of uniqueness, note that any series of the form n≥0 anX is a logarithm for the formal additive group in characteristic p > 0. 6.1. FORMAL GROUPS 179

Suppose h : F → G is a morphism of formal group laws. Diﬀerentiating G(h(X), h(Y )) = h(F (X,Y )) with respect to Y yields

0 0 G2(h(X), h(Y ))h (Y ) = h (F (X,Y ))F2(X,Y ).

Setting Y = 0 and rearranging then gives 1 h0(X) = . F2(X, 0) G2(h(X), 0) and integrating yields the equality

`F (X) = `G(h(X)). (6.1.2.6)

6.1.3 The Functional Equation Lemma. Hazewinkel’s functional equation lemma is a powerful tool for constructing formal group laws with speciﬁed properties, but it can also be used for proving a variety of congruences and integrality statements. We refer the reader to [8] for some of the more exotic applications. Here we use it to construct the Lubin-Tate formal groups. The setup of the functional equation lemma is the following: K is a commutative ring, R ⊆ L is a subring and a ⊆ R is an ideal. Furthermore p is a prime, q is a power of p and we assume that p ∈ a, so that R/a is a ring of characteristic p. Finally there is an endomorphism σ : L → L stabilizing R, such that the induced action of σ on R/a is the qth power Frobenius. For any P n power series f(X) = n>0 bnX ∈ L[[X]] (note that there is no constant term) we deﬁne

X n X qn σ(f) = σ(bn)X , ϕ(f) = σ(bn)X . n>0 n>0

r The last ingredient is a sequence si ∈ L for i ≥ 1 such that σ (si)a ⊆ R for all i ≥ 1 and r ≥ 0. Given f ∈ K[[X]], the functional equation condition relative to (si) is the assertion

X i f(X) − siϕ (f) ∈ R[[X]]. (6.1.3.1) i>0 The functional equation lemma is the content of the next three propositions.

P n 6.1.3.1 Proposition Suppose f = n>0 bnX ∈ K[[X]] satisﬁes the func- × tional equation condition 6.1.3.1, and b1 ∈ R . Then

F (X,Y ) = f −1(f(X) + f(Y )) ∈ R[[X,Y ]].

6.1.3.2 Proposition Suppose f(X) ∈ K[[X]] satisfies the hypotheses of proposition 6.1.3.1. (i) If f¯(X) ∈ K[[X]] is any power series satisfying f¯(0) = 0, 0 −1 ¯ P n f (0) ∈ R and 6.1.3.1 then f (f(X)) ∈ R[[X]]. (ii) If f = n>0 bnX ∈ K[[X]] satisfies 6.1.3.1, and h(X) ∈ R[[X]] then f(h(X)) satisfies 6.1.3.1 as well. 180 CHAPTER 6. THE EXISTENCE THEOREM

6.1.3.3 Proposition Suppose f satisﬁes 6.1.3.1 and g(X), h(X) ∈ R[[X]]. For any r > 0, g(X) ≡ h(X) mod ar if and only if f(g(X)) ≡ f(h(X)) mod ar.

We will give the proofs later in this section. For now we note some conse- quences.

6.1.3.4 Corollary (i) If f ∈ K[[X]] satisfies 6.1.3.1 and f(X) ≡ X mod (X2) then F (X,Y ) = f −1(f(X) + f(Y )) is a formal group law with logarithm f(X). (ii) If F ∈ R[[X,Y ]] is a formal group law whose logarithm satisfies 6.1.3.1, a formal group law G ∈ R[[X,Y ]] is strictly isomorphic to F if and only if its logarithm satisfies 6.1.3.1 with the same sequence (si).

Proof. Assertion (i) is clear. If h is a strict isomorphism F → G and `F , `G are the logarithms of F and G then `G(h(X)) = `F (X). The corollary then follows from propositions 6.1.3.1 and 6.1.3.2. As another example, let

n X Xp X Xn `(X) = , log(1 + X) = (−1)n+1 . pn n n>0 n>0

A quick calculation shows that

1 1 `(X) − `(Xp) = X, log(X) − log(Xp) ∈ [[X]] p p Z(p) and it follows from proposition 6.1.3.2 that the power series

n ! X Xp AH(X) = exp (6.1.3.2) pn n>0 has coeﬃcients in Z(p)[[X]]; it is known as the Artin-Hasse exponential. A more general result along this line is a famous lemma of Dwork. In this case we take K to be the completion of the maximal unramiﬁed extension of Qp, R is its integer ring, and σ is the lifting of the pth power Frobenius to R and K. Dwork’s lemma states that a power series h(X) ∈ K[[X]] with constant term 1 lies in R[[X]] if and only if σ(h(Xp))/h(X)p ∈ 1 + pXR[[X]]. In fact propostion 6.1.3.2 shows that h ∈ R[[X]] if and only if

1 log(h) − ϕ(h) ∈ R[[X]] p and this condition is equivalent to ϕ(h)/hp ∈ 1 + pXR[[X]], as one sees by multiplying by −p and exponentiating. The rest of this section is devoted to the proof of propositions 6.1.3.1-6.1.3.3 We begin with some lemmas. 6.1. FORMAL GROUPS 181

P n × 6.1.3.5 Lemma Suppose f(X) = n>0 anX satisﬁes 6.1.3.1 and a1 ∈ R If r r we write n = q m with q 6 |m then ana ⊆ R. Proof. We argue by induction on n, the case n = 1 being evident. By 6.1.3.1 we can write X i f(X) = g(X) + siϕ (f) i>0 P n with g ∈ R[[X]]. If g = n>0 bnX then

2 r an = bn + s1σ(an/q) + s2σ (an/q2 ) + ··· + srσ (an/qr ).

r The induction step follows on multiplying by a since sia ∈ R.

6.1.3.6 Lemma Suppose f(X) ∈ K[[X]] satisﬁes 6.1.3.1 and g ∈ R[[X1,...,Xd]]. Then i i ϕ(f) ◦ g ≡ ϕ (f ◦ g) mod aR[[X1,...,Xd]] for all i > 0.

P n Proof. If we write f = n>0 anX then

i X i nqi ϕ (f)(g(X·) = σ (an)g(X·) . n>0

Let n = qrm. Since σ is a lift of the qth power Frobenius,

qi i qi g(X·) ≡ σ (g)(X· ) mod a and a straightforward induction shows that

qi+r i qi qr r+1 g(X·) ≡ (σ (g)(X· )) mod a .

Raising the the mth power then yields

nqi i qi n r+1 g(X·) ≡ (σ (g)(X· )) mod a .

r r By lemma 6.1.3.5 we know that if n = q m with q 6 |m then ana ∈ R. For such n we have i nqi i qi n σ (an)g(X·) ≡ (σ (ang)(X· )) mod a and summing over n yields

i X i i qi n ϕ (f)(g(X·)) ≡ σ (an)σ (g)(X· ) mod a n>0 which is ϕ(f)i ◦ g ≡ ϕi(f ◦ g) mod a. 182 CHAPTER 6. THE EXISTENCE THEOREM

P n Proof of proposition 6.1.3.1: The hypothesis is that f = n>0 bnX ∈ K[[X]] × −1 satisﬁes 6.1.3.1 and b1 ∈ R , and in particular f (X) is deﬁned. Write

−1 X F (X,Y ) = f (f(X) + f(Y )) = Fn(X,Y ) n>0

2 × with Fn homogenous of degree n. Since f(X) ≡ b1X mod (X ) and b1 ∈ R , F1(X,Y ) = X + Y ∈ R[X,Y ]. Suppose we have shown that F1,...,Fn−1 ∈ R[X,Y ]. Since F has no constant term,

r r n+1 (F1 + ··· + Fn−1) ≡ F (X,Y ) mod (X,Y ) for all r > 1. Since q > 1,

i i n+1 siϕ (f)(F1 + ··· + Fn−1) ≡ siϕ (f)(F (X,Y )) mod (X,Y ) for all i > 0. Applying lemma 6.1.3.6 and recalling that sia ⊆ R, we ﬁnd

i i n+1 siϕ (f)(F ) ≡ siϕ (f ◦ F ) mod (R[[X,Y ]] + (X,Y ) ) (6.1.3.3) for i > 0. As before we deﬁne g(X) ∈ R[[X]] by

X i f(X) = g(X) + siϕ (f)(X). i>0

Substituting F (X,Y ) for X and applying 6.1.3.3 yields

X i f(F (X,Y )) ≡ g(F (X,Y )) + siϕ (f ◦ F )(X,Y ) i>0 X i ≡ g(F (X,Y )) + siϕ (f(X) + f(Y )) i>0 ≡ g(F (X,Y ) + f(X) + f(Y ) − g(X) − g(Y ) where the congruences are modulo R[[X,Y ]] + (X,Y )n+1. Since g ∈ R[[X]] and f(F (X,Y )) = f(X) + f(Y ),

g(F (X,Y )) ≡ 0 mod (R[[X,Y ]] + (X,Y )n+1).

2 × and since g(X) ≡ b1X mod (X ) and b1 ∈ R , this shows that

F (X,Y ) ≡ 0 mod (R[[X,Y ]] + (X,Y )n+1).

Consequently Fn(X,Y ) ∈ R[X,Y ] as required.

Proof of proposition 6.1.3.2: The proof of part (i) is similar to that of proposition 6.1.3.1 and we will just sketch it. Set f −1(f¯(X)) = h(X), so that f¯ = f ◦ h. 6.1. FORMAL GROUPS 183

P n Then h = n>0 hnX and we know that h1 ∈ R. Suppose we have shown h1, . . . , hn−1 ∈ R. The same argument as before shows that

i i n+1 siϕ (f)(h) ≡ siϕ (f ◦ h) mod (R[[X]] + (X ). where the equivalence is modulo R[[X]] + (Xn+1). Then

X i f(h(X)) ≡ g(h(X)) + siϕ (f ◦ h)(X) i>0 X i ¯ n+1 ≡ g(h(X)) + siϕ (f(X)) mod R[[X]] + X K[[X]] i>0 ≡ g(h(X)) + f¯(X) − g(X) mod R[[X]] + Xn+1K[[X]]

and as before we ﬁnd g(h(X)) ≡ 0, then h(X) ≡ 0, and ﬁnally hn ∈ R. For part (ii) we set f¯ = f ◦ h. With the notation of the previous paragraph we have h ≡ 0 and thus

¯ X i ¯ X i f(X) − siϕ (f)(X) = f(h(X)) − siϕ (f ◦ h)(X) i>0 i>0 X i ≡ f(h(X)) − siϕ (f)(h(X)) i>0 ≡ g(h(X)) ≡ 0.

Proof of proposition 6.1.3.3: Suppose ﬁrst that g(X) ≡ h(X) mod f r. As P n s s before write f(X) = n>0 bnX , so that if n = q m with q 6 |m then bna ∈ R by lemma 6.1.3.5. The same computation as in the proof of lemma 6.1.3.6 shows that s s g(X)q ≡ h(X)q mod ar+s and thus g(X)n ≡ h(X)n mod ar+s for this n. Applying 6.1.3.5 once again then yields

f(g(X)) ≡ f(h(X)) mod ar.

From this we can deduce the more general assertion that g(X) ≡ h(X) mod ar+XnK[[X]] =⇒ f(g(X)) ≡ f(h(X)) mod ar+XnK[[X]]. for any n > 1. In fact if we replace g(X), h(X) by power series g¯(X), h¯(X) with the same terms of degree less than n, and all terms of higher degree equal to zero then f(g(X)) ≡ f(¯g(X)) ≡ f(h¯(X) ≡ f(h(X)) 184 CHAPTER 6. THE EXISTENCE THEOREM where all congruences are modulo ar + XnK[[X]]. Then g¯(X) ≡ h¯(X) mod ar and the assertion follows. To prove the converse implication we ﬁrst show that if h(X) ∈ arR[[X]] then f −1(h(X)) ≡ 0 mod ar. As before, deﬁne g(X) ∈ R[[X]] by f = g + P i ¯ −1 P n −1 i>0 siϕ (f) and write h(X) = f (h(X)) = n>0 cnX . Since f, f and g −1 have no constant terms and their linear terms are b1X, b1 X and b1X respec- r r tively we must have c1 ∈ a . Suppose we have shown ci ∈ a for i ≤ n. Once s again write n = qsm with q 6 |m; then h¯(X)q ≡ 0 mod ar+1 +Xn+1K[[X]] and s therefore f(h¯(X)q ) ≡ 0 mod ar+1 + Xn+1K[[X]] by the result of the previous paragraph. From this we deduce that

¯ ¯ X i ¯ r n+1 h ≡ f ◦ h ≡ g ◦ h + si(ϕ f) ◦ h ≡ 0 mod a + X K[[X]]. i>0

¯ r n+1 2 This shows that g(h(X)) ≡ a + X K[[X]], and since g(X) ≡ b1X mod x , h¯(X) = f −1(h(X)) ≡ ar as required. Suppose now g(X), h(X) ∈ R[[X]] and f(g(X)) ≡ f(h(X)) mod ar. If we set j(X) = f −1(f(h(X)) − f(g(X))) then j(X) ≡ 0 mod ar by what we have just shown. Then

f(h(X)) = f(j(X)) + f(g(X)) or h(X) = F (j(X), g(X)) and consequently

h(X) ≡ F (j(X), g(X)) ≡ F (0, g(X)) ≡ g(X) mod ar which was to be shown.

6.2 Lubin-Tate Groups

6.2.1 Construction of Lubin-Tate formal groups. Suppose now K is a local field with integer ring OK . Let π be a uniformizer of OK , and denote by p and q the characteristic and the cardinality of the residue field of OK .A Lubin-Tate logarithm for π is a power series `(X) ∈ K[[X]] such that 1 `(X) ≡ X mod (X2), `(X) − `(Xq) ∈ O [[X]] (6.2.1.1) π K or in other words, `(X) satisfies the functional equation condition 6.1.3.1 with −1 s1 = π and si = 0 for i > 1. Such power series exist in abundance; for 2 example if g(X) ∈ OK [[X]] is any power series such that g(X) ≡ X mod (X ), the equation 1 `(X) = g(X) + `(Xq) (6.2.1.2) π 6.2. LUBIN-TATE GROUPS 185 can be solved by recursion, yielding an ` ∈ K[[X]] satisyfing 6.2.1.1. Thus if g(X) = X, 2 Xq Xq `(X) = X + + + ··· (6.2.1.3) π π2 For any `(X) satisfying 6.2.1.1 the functional equation lemma is applicable with K, q as above and σ equal to the identity on K. By corollary 6.1.3.4,

−1 F`(X,Y ) = ` (`(X) + `(Y )) (6.2.1.4) is a one-dimensional formal group law over OK , and any two ` satisfying `(X) ≡ X mod (X2) and 6.2.1.1 give rise to strictly isomorphic formal group laws. We will call them Lubin-Tate formal groups associated to π. If the speciﬁc choice of ` does not matter, we denote any of these formal group laws by Fπ. For any a ∈ OK the power series

[a](X) = `−1(a`(X)). (6.2.1.5) has coeﬃcients in OK by proposition 6.1.3.1. It is immediate that the [a](X) is 2 an endomorphism of the formal group law F`. Since `(X) ≡ X mod (X ) we must have [a](X) ≡ aX mod (X2), and this shows that a 7→ [a] is an injective ring homomorphism OK → End(F`). Observe that [a] is an automorphism of F` if and only if a is a unit in OK . Since

`([a](X)) = a`(X) (6.2.1.6) by deﬁnition, we see that [a](X) is the nth power endomorphism when a = n is a natural number. Let `(X) be a Lubin-Tate logarithm and deﬁne g(X) by 6.2.1.2. Setting a = π in 6.2.1.6 and multiplying by π yields

`([π](X) = π`(X) = πg(X) + `(Xq) ≡ `(Xq) mod (π).

Proposition 6.1.3.3 then implies that

[π](X) ≡ Xq mod (π) (6.2.1.7) and on the other hand the deﬁnition 6.2.1.5 shows that

[π](X) ≡ πX mod (X2). (6.2.1.8)

The next proposition and its corollary will show that the logarithm ` can be recovered from the series [π], and that in fact any power series in OK [[X]] satisfying 6.2.1.7 and 6.2.1.8 is [π](X) for a unique Lubin-Tate logarithm. We will see later that it is useful to use a logarithm for which the endomorphism [π](X) has a given form, e.g. a polynomial of degree q. In fact the original development of the theory starts with a power series g(X) satisfying 6.2.1.7 and 6.2.1.8 and constructs the formal group law from g(X). 186 CHAPTER 6. THE EXISTENCE THEOREM

6.2.1.1 Proposition For any g(X) ∈ R[[X]] such that

g(X) ≡ πX mod (X2), g(X) ≡ Xq mod (π) (6.2.1.9) the limit 1 `(X) = lim g◦n(X) (6.2.1.10) n→∞ πn exists and is a Lubin-Tate logarithm. The associated formal group law satisﬁes [π](X) = g(X). Proof. Once we show that the limit is a Lubin-Tate logarithm the last statement is clear since 1 `(g(X)) = lim g◦(n+1)(X) = π`(X) = `([π](X)). n→∞ πn For n ≥ 1 set 1 g (X) = g◦n(X). n πn 2 Clearly gn(X) ≡ X mod (X ), so the rest of the proposition follows if we show that

q n−1 n−1 gn+1(X) ≡ gn(X) mod (π, X ) + X K[[X]] (6.2.1.11) 1 g (X) − g (Xq) ∈ R[[X]] (6.2.1.12) n+1 π n n q n for all n ≥ 1. We ﬁrst show that π gn(X) ∈ (π, X ) . For n = 1 this follows from 6.2.1.9 and if this has been proven for n then

n+1 n q n q qn q n+1 π gn+1(X) = g(π gn(X)) ∈ π(π, X ) + (π, X ) ⊆ (π, X ) . If we write

g(X) = πX + h(X) + Xq with h(X) ∈ πX2R[[X]]. then 1 g (X) = g(πng (X)) n+1 πn+1 n 1 = (π · πng (X) + h(πng (X)) + (πng (X))q) πn+1 n n n h(πng (X)) (πng (X))q = g (X) + n + n . n πn+1 πn+1 Since h(X) ∈ πX2R[[X]], h(πng (X)) 1 n ∈ (π, Xq)2n ⊆ (π, Xq)n + XqnK[[X]] πn+1 πn and similarly (πng (X))q 1 n ∈ (π, Xq)qn ⊆ (π, Xq)n(q−1)−1 + Xn(q−1)−1K[[X]]. πn+1 πn+1 6.2. LUBIN-TATE GROUPS 187

The congruence 6.2.1.11 follows since nq > n(q − 1) − 1 ≥ n − 1. The congruence 6.2.1.9 implies

g(X + πnY ) ≡ g(X) mod πn+1R[[X,Y ]] and by induction we ﬁnd

g◦n(X + πY ) ≡ g(X) mod πn+1R[[X,Y ]] for all n ≥ 1. If we write g(X) = πh(X) + Xq and substitute Xq and h(X) for X and Y we ﬁnd that

g◦(n+1)(X) = g◦n(Xq + πh(X)) ≡ g◦n(Xq) mod πn+1 and 6.2.1.11 follows on dividing by πn+1.

6.2.1.2 Corollary The logarithm `(X) of a Lubin-Tate group associated to π satisﬁes 1 `(X) = lim [πn](X). (6.2.1.13) n→∞ πn Proof. This follows from 6.2.1.7–6.2.1.8 and [π]◦n = [πn].

Since F` depends only on π up to strict isomorphism we can denote it by Fπ, if we do not need to specify the logarithm `. On the other hand these formal groups really do depend on the choice of π, at least over OK . Nontheless:

0 0 6.2.1.3 Proposition Suppose π, π are uniformizers of OK and π = uπ, so × 0 0 that u ∈ OK . If ` and ` are Lubin-Tate logarithms attached to π and π , there is a power series h(X) ∈ O [[X]] and v ∈ O× such that σ−1v = u and Kˆ nr Kˆ nr

`0 ◦ h = v`, (6.2.1.14) h(X) ≡ vX mod (X2) (6.2.1.15) σh = h ◦ [u]. (6.2.1.16)

Proof. The existence of v ∈ O× satisfying σ−1v = u follows from Lemma Kˆ nr 1.3.6.2. Then 1 1 v` − ϕ(v`) = v` − σvϕ(`) π0 π0 1 1 = v` − vϕ(`) = v(` − ϕ(`)) ∈ O [[X]] π π Kˆ nr and if we set h(X) = (`0)−1(v`(X)) 188 CHAPTER 6. THE EXISTENCE THEOREM

proposition 6.1.3.1 shows h(X) ∈ OKˆ nr [[X]]. Furthermore 6.2.1.14 and 6.2.1.15 follow by construction, and

σh(X) = (`0)−1(σv`(X)) = (`0)−1(vu`(X)) = (`0)−1(v`([u](X))) = h([u](X)) establishes 6.2.1.16.

0 6.2.1.4 Corollary If π and π are any uniformizers of OK , the formal group nr laws Fπ, Fπ0 are isomorphic over the integer ring of the completion Kˆ of the maximal unramiﬁed extension of K.

Proof. In fact 6.2.1.14 shows that h is a morphism Fπ → Fπ0 , and since v is a unit 6.2.1.15 shows that it is an isomorphism. A Lubin-Tate logarithm and its inverse have convergence properties analogous to the usual logarithm and exponential.

6.2.1.5 Proposition Suppose `(X) is a Lubin-Tate logarithm and F` is the P n associated formal group law. (i) The power series `(X) = n>0 anX converges for |X| < 1. (ii). Denote by e(X) the functional inverse of `(X), and for any extension L/K of nonarchimdean ﬁelds denote by DL the open disk

−1 DL = {x ∈ OL | v(x) > (q − 1) } where v is the valuation of L such that v(π) = 1. Then e(X) converges on DL and `(X) and e(X) give inverse isomorphism of F`(L) with the additive group of DL.

r r Proof. If n = q m with q 6 |m then lemma 6.1.3.5 shows that π an ∈ R, so for any x ∈ OL,

n v(anx ) = v(an) + nv(x) > nv(x) − logq(n) → ∞ as n → ∞

r which proves (i). If x ∈ DL and n = q m as before, r r q − 1 v(a xn) ≥ v(a xq −1) + v(x) ≥ − r + v(x) ≥ v(x) n n q − 1 which shows that ` maps DL to itself, and that v(`(x)) ≥ v(x) for x ∈ DL. Since e(X) is a formal inverse to ` it suﬃces for (ii) to show that e(X) converges on DL and maps DL to itself. When K has characteristic p > 0, the logarithm 6.2.1.3 clearly satisﬁes

`(X + Y ) = `(X) + `(Y ). (6.2.1.17) 6.2. LUBIN-TATE GROUPS 189

Since the formal group law is determined by `, the Lubin-Tate formal group ˆ associated to ` is just the formal additive group Ga. In this case the group itself is not of so much interest and one speaks instead of a formal OK -module ˆ structure on Ga, or simply a formal OK -module. The interest of this case is that this notion can be globalized, i.e. K can be replaced by the function ring of a smooth affine curve over a finite field. This leads to the notion of a Drinfeld module, which plays a central role in the global Langlands correspondence for function fields.

6.2.2 Division points. Fix a Lubin-Tate group Fπ associated to a uniformizer π of K. The division points of Fπ are the torsion points of Fπ(m) × considered as an OK -module. Since [a] is an isomorphism for a ∈ OK , the group of division points is the union of the kernels of the endomorphisms [πn] n on Fπ(m). We denote by An the kernel of [π ] on F (m). Since any two such groups are strictly isomorphic over OK the OK -module Fπ(m) is independent of the particular choice of Fπ up to isomorphism (but not independent of π). The same then follows for the groups An. Finally, since the coeﬃcients of the series [π](X), [a](X) are in K, the absolute Galois group GK of K acts on An, and this action is linear for the OK -module structure of An. Finally, for every n the inclusion An → An+1 is OK -linear and Galois-equivariant, as is the map An+1 → An induced by [π]. Using these maps we deﬁne A = lim A T (F ) = lim A . (6.2.2.1) ∞ −→ n π ←− n n n

The group T (Fπ) called the Tate module of Fπ.

n 6.2.2.1 Lemma For all n, An is isomorphic to OK /π OK as an OK -module. Furthermore A∞ is isomorphic to K/OK and T (Fπ) is isomorphic to OK as OK -modules.

Proof. Since An is a ﬁnite group it is a ﬁnitely generated OK , and therefore a sum of cyclic OK -modules since OK is a PID. By proposition 6.2.1.1 we can choose Fπ such that [π](X) is a polynomial g(X) = πX + ··· + Xq (6.2.2.2) of degree q. Since g is separable it has q roots, which means that A1 has order q. This implies that A1 'OK /πOK as an OK -module, and it follows that An as well. Since n g◦n(X) = πnX + ··· + Xq n n An has cardinality q , so we must have An 'OK /π OK . The remaining statements are immediate.

6.2.2.2 Corollary There is a homomorphism

× κ : GK → OK (6.2.2.3) 190 CHAPTER 6. THE EXISTENCE THEOREM such that s(x) = [κ(s)](x) (6.2.2.4) for all x ∈ A∞. × Proof. If u ∈ An generates An as an OK -module there is a κn(s) ∈ OK such that s(u) = [κn(s)](u), and then s(x) = [κn(s)](x) for all x ∈ An. Since κn(s) is well-deﬁned modulo πn we must have

n κn+1(s) ≡ κn(s) mod π for all n, and κ(s) = lim κ (s) satisﬁes 6.2.2.4. ←−n n We next deﬁne

n [ n Kπ = K(An),Kπ = K(A∞) = Kπ . (6.2.2.5) n

n Note that Kπ is generated by any generator of the OK -module. Thus if we assume, as we may that [π](X) = πX + ··· + Xq is a monic polynomial of n degree q, Kπ is the splitting ﬁeld over K of the polynomial [πn](X) [π]([πn−1](X)) f (X) = = = g([πn−1](X)) n [πn−1](X) [πn−1](X) (6.2.2.6) [π](X) where g(X) = = Xq−1 + ··· + π. X

Note also that g(X) is an Eisenstein polynomial of degree q − 1, so fn(X) is an Eisenstein polynomial of degree (q − 1)qn−1.

6.2.2.3 Theorem The homomorphism 6.2.2.3 factors through an isomorphism

∼ × κπ : Gal(Kπ/K) −→OK . (6.2.2.7) Proof. By construction κ factors through a homomorphism

× κπ : Gal(Kπ/K) → OK . × The groups Gal(Kπ/K), OK have filtrations n × n × Gn = Gal(Kπ /K) ⊂ Gal(Kπ/K),Un = Ker(OK → (OK /π ) ) and by construction κ (G ) ⊆ U . Since G ' lim G and O× ' lim U it π n n ←−n n K ←−n N ∼ suffices to show that κπ induces isomorphisms Gn −→ Un for all n. The question is independent of the particular choice of Lubin-Tate formal group, so we may assume that [π](X) has the form assumed in 6.2.2.6. Evidently Gn → Un is injective, for if s ∈ Gn maps to u ∈ Un it fixes every element of An. Thus it suffices to show that |Gn| = |Un| for all n. In n−1 fact |Un| = (q − 1)q , and since fn is an Eisenstein polynomial of degree n−1 n n−1 (q − 1)q , |Gn| = [Kπ : K] = (q − 1)q . 6.3. THE LUBIN-TATE RECIPROCITY LAW 191

n 6.2.2.4 Proposition The uniformizer π is a norm from Kπ for all n > 0.

Proof. With the notation of 6.2.2.6 and the preceding proof, if u ∈ An \ An−1 then N n (−u) = π. Kπ /K

6.3 The Lubin-Tate Reciprocity Law

6.3.1 Proof of the existence theorem. The existence theorem will be nr proven by showing that K Kπ is the maximal abelian extension of K for any uniformizer π, and then relating the norm residue symbol to the homomorphism κ deﬁned above. nr By proposition 1.2.5.2 K and Kπ are linearly disjoint over K. This yields isomorphisms nr nr K Kπ ' K ⊗K Kπ (6.3.1.1) and nr nr Gal(K Kπ/K) ' Gal(K /K) × Gal(Kπ/K). (6.3.1.2)

nr From this it follows that the inertia group Iπ ⊂ Gal(K Kπ/K) can be identiﬁed with Gal(Kπ/K).

6.3.1.1 Lemma Suppose K is a nonarchimedean ﬁeld, L/K is an algebraic extension and let Lˆ be the completion of L for the extension of the valuation of K to L. If x ∈ Lˆ is separable algebraic of L then x ∈ L.

Proof. Let Lsep be a separable closure of L. We can identify the separable closure L0 of L in Lˆ with a subfield of Lsep , so it suffices to show that L0 = L. Equivalently we must show that Gal(Lsep /L) = Gal(Lsep /L0), but this is clear: by continuity, an automorphism of Lsep fixing L must by continuity fix L0 as well, and conversely an automorphism fixing L0 must fix L.

6.3.1.2 Proposition Let K be a local ﬁeld and let Knr be its maximal unram- 0 nr nr iﬁed extension. For any two uniformizers π, π of OK , K Kπ = K Kπ0 .

Proof. We take all fields in the proposition to be subfields of the completion of a fixed algebraic closure of K. By proposition 6.2.1.3 the formal group laws nr nr Fπ, Fπ0 are isomorphic over the integer ring of K . This shows that Kˆ Kπ = nr nr nr Kˆ Kπ0 . It follows that K Kπ and K Kπ0 have the same completion, and nr nr then 6.3.1.1 shows that K Kπ = K Kπ0 .

nr For the rest of this section we will write L = K Kπ since this is independent × of π. If π is a uniformizer of OK , any element of K has a unique expression n × × as π u with u ∈ OK and we can deﬁne a homomorphism rπ : K → Gal(L/K) 192 CHAPTER 6. THE EXISTENCE THEOREM by setting ( nr σarith on K rπ(π) = 1 on Kπ (6.3.1.3) ( nr 1 on K × rπ(u) = −1 −1 for u ∈ OK . κπ (u ) on Kπ

× 6.3.1.3 Lemma The homomorphism rπ : Gal(L/K) → K is independent of π

Proof. Since K× is generated as a group by uniformizers it suﬃces to show 0 0 0 that rπ(π ) = rπ0 (π ) for any two uniformizers π, π of K. Since since both are nr 0 σarith on K it suﬃces to show they are equal on Kπ0 , and since rπ0 (π ) is the 0 identity on Kπ0 we must show that the same is true for rπ(π ). 0 0 0 Let ` and ` be logarithms associated to π and π and let A∞ and A∞ be the groups of division points of the associated formal groups. The isomorphism 0 A∞ → A∞ is given by the power series h(X) in proposition 6.2.1.3, so with the s above notation we must show that h(x) = h(x). Since h ∈ OKˆ nr [[X]], s induces nr −1 σ on Kˆ , while on A∞ it induces [u ]. Then 6.2.1.16 shows that

sh(x) = sh(sx) = σh([u−1](x)) = (h ◦ [u])([u−1](x)) = h(x) as required. We can now compute the norm residue symbol on the maximal abelian extension Kab of K.

6.3.1.4 Theorem Let K be a local ﬁeld with uniformizer π. (i) The ﬁeld nr K Kπ is the maximal abelian extension of K. (ii) Let π be a uniformizer × n × of K and for all x ∈ K write x = π u with u ∈ OK . Then ( σn(a) a ∈ Knr θ(x)(a) = −1 (6.3.1.4) [u ](a) a ∈ Kπ.

Proof. Denote by r : K× → Gal(L/K) any of the “common value” of the maps × rπ. Since K is generated by uniformizers, the correctness of 6.3.1.4 for a ∈ L will follow if we show that θ(π) = r(π) for any uniformizer π. First, θ(π) and nr r(π) both induce σ on K , and 6.3.1.3 shows that r(π) is the identity on Kπ. n On the other hand π is a norm from Kπ for all n, so θ(π) is the identity on Kπ as well. Therefore θ(x)|L = r(x) (6.3.1.5)

ab which proves (ii). If Iπ ⊂ Gal(L/K), I ⊂ Gal(K /K) are the inertia subgroups 6.3.1.4 shows that the composite map

× θ κπ × OK −→ I → Iπ −−→OK 6.3. THE LUBIN-TATE RECIPROCITY LAW 193 is u 7→ u−1. It is therefore an isomorphism, and as the ﬁrst two maps are surjections and the last is an isomorphism, the canonical projection I → Iπ must be an isomorphism as well. Then Gal(Kab /K) → Gal(L/K) is an isomorphism, which proves (i).

6.3.1.5 Corollary The reciprocity map θ : K× → Gal(Kab /K) induces a topo- × ab logical isomorphism of OK with the inertia subgroup of Gal(K /K).

Proof. This follows from the identiﬁcation I ' Gal(Kπ/K) and 6.3.1.4, which shows that θ is the composition of two continuous bijections. Since all groups involved are compact, a continuous bijection is a homeomorphism. The corollary is the main step in the proof of the existence theorem:

6.3.1.6 Theorem Let K be a local field. (i) The reciprocity map θ : K× → Gal(Kab /K) is a continuous injection with dense image. (ii) The topology induced on K× is the topology defined by the open subgroups of finite index. (iii) For every open subgroup U ⊆ K× of finite index there is a finite abelian exten- × sion L/K such that NL/K (L ) = U. Proof. The topology of K× is the product topology for any isomorphism K× ' × OK ×Z determined by a choice of uniformizer. That θ is continuous then follows from corollary 6.3.1.5. That it is an injection with dense image follows from the commutative diagram

× × 0 / OK / K / Z / 0

θ θ 0 / I / Gal(Kab /K) / Zˆ / 0 in which the rows are exact, the left vertical arrow is an isomorphism and the right vertical arrow is an injection with dense image. This establishes (i), and (ii) follows from (i). Finally (iii) follows from (ii) since an open subgroup of ﬁnite index in K× containing a norm group is itself a norm group.

6.3.2 Cyclotomic Extensions. As a ﬁrst application of these results we have the local Kronecker-Weber theorem:

6.3.2.1 Theorem Any ﬁnite abelian extension of Qp is a subﬁeld of Qp(µn) for some n.

Proof. The ordinary logarithm function is a Lubin-Tate logarithm for Qp, and ˆ ˆ the associated formal group is Gm. The division points of Gm are the values 1 − ζ where ζ runs through al p-power roots of unity. Finally any unramiﬁed extension of Qp is obtained by adjoining to Qp roots of unity of order prime to p. It follows from (i) of theorem 6.3.1.4 that the maximal abelian of extension of 194 CHAPTER 6. THE EXISTENCE THEOREM

¯ Qp is obtained by adjoining all roots of unity in Qp to Qp, and this establishes the theorem.

Shafarevich noticed that the global Kronecker-Weber theorem is a simple consequence of the local theorem 6.3.2.1. The proof needs just one nontrivial global result, a well-known theorem of Minkowski:

6.3.2.2 Theorem If K/Q is a ﬁnite unramiﬁed extension then K = Q.

The proof can be found in any of the standard references, e.g. [11, Ch. 4].

6.3.2.3 Theorem Any ﬁnite abelian extension of Qp is a subﬁeld of Q(µn) for some n.

Proof. Suppose K/Q is abelian, and for all primes p let Kp be the completion of K at some place p of K above p. By the local Kronecker-Weber theorem 6.3.2.1,

Kp ⊆ Qp(µpnp ) for some integer np. Since K is ramified at finitely many primes, np is prime to ep p for all but finitely many p. Write np = p mp with p 6 |mp. Then ep = 1 for almost all p and we set Y n = pep . p

We will show that K ⊆ Q(µn). The field L = K(µn) is an abelian extension of Q. If we denote by Lp the completion of L at some place above the prime p of K, then L ' K (µ e 0 ) for p p p np 0 some np not divisible by p. Thus p is ramified in K if and only if it is ramified in L. The inertia group Ip of L above p coincides with the inertia group of Lp/Qp, ep so |Ip| = φ(p ). Let I ⊆ Gal(L/Q) be the subgroup generated by the Ip. Since Gal(L/Q) is abelian,

Y Y np |I| ≤ |Ip| = φ(p ) = φ(n) = [Q(µn): Q]. p p

The fixed field LI of I is unramified above Q at all primes p, so by LI = Q by Minkowski’s theorem 6.3.2.2. Then I = Gal(L/Q), and the last inequality implies that

[L : Q] = |I| ≤ [Q(µn): Q].

Since Q(µn) ⊆ L we conclude that L = Q(µn), and then K ⊆ Q(µn). Shafarevich also gave an elementary if somewhat lengthy proof of the local theorem 6.3.2.1. A somewhat simplied proof of this was given by Narkiewicz; an account can be found in [15]. 6.3. THE LUBIN-TATE RECIPROCITY LAW 195

6.3.3 The ramification filtration. We first compute the ramification fil- n tration and the Herbrand function for the extensions Kπ /K; the corresponding ab filtration for Kπ/K and K /K follows immediately from this. The computation is a direct generalization of the computation in the cyclotomic case. i × As before we denote by U ⊆ OK the subgroup of units congruent to 1 i i i+1 n modulo m . Suppose u ∈ U \ U and set s = θ(u). If πn ∈ Kπ is any uniformizer then i(s) = v(s(πn) − πn) where v is the normalized valuation of n n Kπ . We may take πn to be a generator of Aπ as an OK -module, so that n n−1 [π ](πn) = 0 and [π ](πn) 6= 0. By theorem 6.3.1.4 we have

−1 s(πn) = [u ](πn).

−1 i × We can write u = 1 + π v with v ∈ OK ; then

i i s(πn) = [1 + π v](πn) = Fπ(πn, [π v](πn)) (6.3.3.1)

i n−i n−i−1 where Fπ is the group law. Now β = [π v](πn) lies in A \ A , and is n−i n n−i i thus a uniformizer of the ﬁeld Kπ . Since Kπ /Kπ has degree q we must have v(β) = qi. If we write

X i j Fπ(X,Y ) = X + Y + cijX Y i,j>1 then 6.3.3.1 says that

X i j s(πn) = πn + β + cijπnβ . i,j>1

Since cij ∈ OK and πn, β have positive valuation,

i v(s(πn) − πn) = v(β) = q and therefore i i−1 i Gu = θ(U ) for q − 1 < u ≤ q − 1. (6.3.3.2) The computation of the Herbrand function is just like the cyclotomic case. For u ∈ (qi−1 − 1, qi − 1) we have

0 × i 1 ϕKn/K (u) = [O : U ] = π K (q − 1)qi−1 and so Gv = θ(U i) for i − 1 < v ≤ i. (6.3.3.3) In the above discussion i ≤ n, but since the upper numbering is compatible ab with quotients, 6.3.3.3 holds for the extension Kπ/K as well. Finally K /K ab and Kπ have the same inertia group, so 6.3.3.3 holds for K . We have thus proven the Hasse-Arf theorem for local fields: for any abelian extension L/K, the breaks of the ramification filtration in the upper numbering are at integers. 196 CHAPTER 6. THE EXISTENCE THEOREM Bibliography

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