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Local Class Field Theory Local Class Field Theory 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 homo- morphisms 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 ab- solute value on R is a map R ! R≥0, written x 7! jxj, with the following properties: jxj = 0 if and only if x = 0: (1.1.1.1) jxyj = jxjjyj: (1.1.1.2) jx + yj ≤ jxj + jyj: (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 j1j = j − 1j = 1, so that j − xj = jxj for all x 2 R. Then 1.1.1.3 is equivalent to the triangle inequality: jx − yj ≤ jxj + jyj. • If x 2 R× then 1.1.1.1 and 1.1.1.2 imply jx−1j = jxj−1. • If R is a division ring and j j is map satisfying 1.1.1.2, then 1.1.1.1 is equivalent the condition that j j 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 jR×j 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: jx + yj ≤ max(jxj; jyj) for all x, y 2 R: (1.1.1.4) An absolute value that is not nonarchimedean is called archimedean. If j j is nonarchimedean, the triangle inequality says that jx−yj ≤ max jxj; jyj. From this we deduce the following sharper form of 1.1.1.3: jxj 6= jyj ) jx + yj = max jxj; jyj: (1.1.1.5) Suppose for example jxj < jyj; if jx + yj < jyj then jyj = jx + y − xj ≤ max(jx + yj; jxj) < jyj, a contradiction. We could rephrase 1.1.1.5 by saying that every triangle is isosceles. For a 2 R and r ≥ 0 the sets B(a; r) = fx 2 R j jx − aj ≤ rg (1.1.1.6) B(a; r−) = fx 2 R j jx − aj < rg (1.1.1.7) (1.1.1.8) are the closed and open ball of radius r about a respectively. Another conse- quence of 1.1.1.5 is that if two balls are not disjoint then one is contained in the other. 1.1.1.1 Lemma Suppose K is a division ring. A function j j : K ! R satis- fying 1.1.1.1 and 1.1.1.2 satisfies 1.1.1.4 if and only if jxj ≤ 1 implies jx+1j ≤ 1 for all x 2 K. Proof. The condition is clearly necessary. If jxj ≤ jyj and y 6= 0 then jxy−1j ≤ 1, so that jxy−1 + 1j ≤ 1 and thus jx + yj ≤ jyj = max(jxj; jyj). 1.1.2 The integer ring and its ideals. If j j is a nonarchimedean valu- ation on R, the set OR = fx 2 R j jxj ≤ 1g 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 = fx 2 K j jxj = 1g: (1.1.2.1) × For example if x 2 K satisfies jxj < 1 then 1 + x 2 OR . For any ring R the sets Ir = fx 2 R j jxj ≤ rg (1.1.2.2) I<r = fx 2 R j jxj < rg are 2-sided ideals of OR. 1.1. ABSOLUTE VALUES 7 1.1.2.1 Proposition Suppose K is a division ring with a nonarchimedean ab- solute value and S ⊆ OK is a nonempty subset. If r = sup jxj: (1.1.2.3) x2S the left (resp. right) ideal of OK generated by S is Ir if there is an x 2 S such that jxj = r, and I<r otherwise. 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 2 I with jxj = r; then y 2 Ir implies −1 −1 jyj ≤ jxj, so that jyx j ≤ 1, yx 2 OK and finally y = (y=x)x 2 I. If there is no x 2 I such that jxj = r we must have I ⊆ I<r, and we can find a sequence of xn 2 I with jxnj ! r. For any y 2 I<r there is an n such that jyj ≤ jxnj and as before we have y ⊆ xnOK ⊆ I and consequently I<r ⊆ I. 1.1.2.2 Corollary Suppose K is a division ring with a nonarchimedean abso- lute 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 jxj = 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 char- acteristic 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 2 K× then jaxa−1j = jxj for all x 2 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 2 K. If the residue field k is commutative, adk(u) 8 CHAPTER 1. NONARCHIMEDEAN FIELDS × is trivial for any u 2 OK , and it follows that adk(a) depends only on jaj. 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 [ f1g defined by ( − log x x 6= 0 v(x) = (1.1.3.1) 1 x = 0 where the logarithm can have any base greater than one. The function v(x) clearly satisfies the conditions v(x) = 1 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 [ f1g satisfying the conditions 1.1.3.2- 1.1.3.4 defines an valuation by taking jxj = a−v(x) for any a > 1. A function R ! R [ f1g satisfying 1.1.3.2-1.1.3.4 is called a valuation.
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