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Home , Brauer group, Galois cohomology, Galois module, Perfect field

Galois and the

February 1, 2007

1 Galois Modules.

Let G be a profinite group. Main Examples: G = GF := Gal(F¯ /F ) where F is a field, and —for these notes only—F¯ will mean a choice of separable of F . Also: G = GK,S for K a number field and S a (finite, usually) set of places of K. Also: or G = the profinite completion of your favorite “discrete” group. We will call a G- M topological if the isotropy subgroup of any element m ∈ M (under the G-action) is an open subgroup (of finite index) in G. By a (over F ) we will mean a topological G-module for G = GF for F a field (note the curious logical issue related to this) and if F = K a number field and G = GK,S a topological G-module is nothing other than a Galois module over K unramified outside S.

Cohomology of topological G-modules are given as direct limits:

Hq(G, M) := lim Hq(G/H, M H ) H the limit over the system of open normal subgroups H ⊂ G.

∗ Main Example for the moment: M = F¯ with natural action of GF . Called, sometimes the .

0 ∗ ∗ Theorem 1 • H (GF , F¯ ) = F

1 ∗ • H (GF , F¯ ) = 0

Proof. HT90.

2 Intro to the Brauer group

2 ∗ Definition 1 The Brauer group of F is Br(F ) := H (GF , F¯ ).

1 Discuss (in)dependence of choice of F¯.

Theorem 2 • If F ⊂ K ⊂ L ⊂ F¯ are fields, with K/F and L/F finite Galois field extensions, then the natural inflation mappings

H2(Gal(K/F ),K∗) → H2(Gal(L/F ),L∗)

are injections.

• [ [ Br(F ) = Br(K/F ) := H2(Gal(K/F ),K∗), K K where the union is taken over the system of finite Galois extensions K/F (or equivalently, of the the system of finite (separable) extensions in

• The natural inflation-restriction sequence gives us an exact sequence

0 → H2(Gal(K/F ),K∗) → Br(F ) → Br(K).

Example: If F is a perfect field of p then Br(F ) has no p-torsion. If K/F is a finite (separable, say) extension of n then Br(K/F ) is killed by n. ( The composition of corestriction and restriction Br(F ) → Br(K) → Br(F ) is multiplication by the degree [K : F ].)

3 When the Brauer group is universally trivial

Let F be a field.

Theorem 3 The following are equivalent.

• For every finite K/F , Br(K) = 0.

• For every finite L/K with K finite a separable over FL∗ is cohomologically trivial as Gal(L/K)-module.

• For every finite Galois extension L/K with K finite a separable over F the norm mapping ∗ ∗ NL/K : L → K is surjective.

Proof. This uses the “two consecutive vanishing dimensions” theorem.

Note: If F is a field with no inseoarable algebraic extensions (e.g., perfect, characteristic 0, . . . ) it is equivalent to say that there is no noncommutative division algebras of finite dimension over F . This, however, uses:

2 4 The Brauer group as group of equivalence classes of Central Simple Algebras (Intro)

Theorem 4 ∗ Let A be a finite dimensional algebra over F . The following are equivalent

1. A possesses no nontrivial two-sided ideals, and the of A is F .

2. The base change of A to the algebraic closure of F is isomorphic to a (finite-dimensional) matrix algebra over the algebraic closure of F .

3. There is a finite Galois extension K/F such that the base change of A to K is isomorphic to a (finite-dimensional) matrix algebra over K.

4. A is isomorphic to a matrix algebra with coefficients in a division Dover F with center equal to F (or equivalently: A ' D ⊗F Mn(F )). 5. (Definition) A is a CSA over F .

Proof. (to be discussed)

The category of CSA’s are closed under (over F ) and the passage from a CSA A to its opposite F -algebra Ao is an involution of the category. The degree of a CSA over F is a perfect square (since it is the degree of a matrix algebra over F ). Define Br0(F ) := the set of equivalence classes of CSA’s defined in the usual way: Two CSA’s over F A, A0 are equivalent if there are positive integers n and n0 and an (of F -algebras)

0 A ⊗F Mn(F ) ' A ⊗F Mn0 (F ).

Note that since Mn(F ) ⊗F Mm(F ) ' Mnm(F ) the relation ' is indeed an equivalence relation, and also ' respects tensor products. Since (A ⊗ B) ⊗ C ' A ⊗ (B ⊗ C) and A ⊗ B ' B ⊗ A, tensor product puts a “commutative structure” on Br0(F ). One sees that Br0(F ) is a group with Ao the inverse of A since:

Lemma 1 The natural F -algebra homomorphism

o A ⊗F A → EndF −(A)

o given by a ⊗F b(x) = axb for a ∈ A; b ∈ A , x ∈ A is an isomorphism.

Theorem 5 There is a natural (functorial) isomorphism Br0(F ) ' Br(F ).

Proof. (to be discussed)

From the previous stated results we see that every element of Br0(F ) is representable by a D over F with center F . Every such division algebra has square rank d2 over F .

3 Definition 2 By the reduced norm of a division algebra over F with center F (and of rank d2) let us mean the homomorphism N˜ : D∗ → F ∗ defined as follows. Find an isomorphism D⊗F F¯ ' Md(F¯) and for x ∈ D put N˜(x) = detF (x⊗F 1).

A priori, detF (x ⊗F 1) lies in F¯ but since it is invariant under the natural action ofGF we have that it lies in F , and since D is a division algebra, it lies in F ∗. Moreover, the mapping N˜ is independent of the way in which we identify D ⊗F F¯ with Md(F¯) since any two ways will differ only by conjugation, and the determinant is insensitive to conjugation. If x1, x2, . . . , xd2 is an F - P basis for the vector space D over F , so that a general vector x ∈ D can be written as x = i aixi we have that ˜ N(x) = P (a1, a2, . . . , ad2 ) 2 for P (t1, t2, . . . , td2 ) some homogeneous polynomial with coefficients in F in d variables of degree d.

Corollary 6 If a field F has the property that its Brauer group is nontrivial, there is integer d > 1 2 and a homogeneous polynomial in d variables P (t1, t2, . . . , td2 ) with coefficients in F and of degree d, and having no nontrivial solutions in F .

Which leads us to:

5 Quasi-algebraically-closed fields

A field F is Quasi-algebraically-closed if every homogeneous polynomial with coefficients in F of degree less than the number of its variables has a nontrivial F -. A wonderful, but completely elementary, theorem of Lang shows that QAC is a ”universal” property (in the sense that if F is QAC then any finite extension of F is QAC as well). A corollary of this and of the previous corollary is that if F is QAC then F has universally trivial Brauer group; i.e., every finite extension K of F has trivial Brauer group.

Examples:

• Finite fields. Chevalley-Warning Theorem. Let F be a finite field of characteristic p and P a (not necessarily homogeneous) polynomial over F having more variables than its degree. Then the number of solutions (in F ) is congruent to 0 modulo p. (Ergo, if P has vanishing constant term, it must have nontrivial solutions over F .) [I’ll give one of the traditional proofs of this. I’ll also give, as alternative, a proof of the weaker statement that F has (universally) trivial Brauer group as in Wedderburn’s Theorem, the proof given, say, at the beginning of Weil’s Basic . Also, for the first homework assignment I’ll ask that people write out some version of a proof that H2(Gal(L/K),L∗) = 0 for L a finite field, using what we know cohomologically. Also, some examples.]

4 unr • Let K be a finite degree extension of Qp and K /K a maximal unramified subextension in K¯ . Then Kunr has trivial Brauer group.

Further Examples

• C[[t]], and more generally Lang’s result that a complete discrete ring with alge- braically closed residue field is quasi-algebraically closed.

• F = R where Br(R) is cyclic of order two, with the as example of division P 2 algebra. Also i xi as homogeneous form. • Tsen’s Theorem. If F is a function field in one variable over an algebraically closed field, then F is quasi-algebraically closed.

• Theorem of Harris, Graber, and Starr If F is a function field in one variable over an algebraically closed field of characteristic zero, then any rationally connected variety over F has an F -rational point.

• Any of Q containing all roots of unity has trivial Brauer group.

6 Nonabelian cohomology and descent

(to be continued)

5