Etale cohomology of fields by Johan M. Commelin, December 5, 2013

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1 Etale cohomology of fields by Johan M. Commelin, December 5, 2013 Etale cohomology The canonical topology on a Grothendieck topos Let E be a Grothendieck topos. The canonical topology T on E is given in terms of coverings by the collection of all families fx i! X g i of universal eective epimorphisms, i.e., for every X 0! X, the family fx 0 i! X 0 g i = fx i z X X 0! X z X X 0 g i makes the diagram Y Y Hom(X 0 ; Z)! Hom(X 0 ; i Z)! Hom(X 0 i z X X 0 ; 0 j Z) exact for all Z 2 E. i Theorem. The Yoneda embedding gives an equivalence of categories EæSh(E; T ). i;j Profinite groups; G-sets A pronite group is by denition a group that is the limit of a diagram of nite groups. Every pronite group is the limit of the diagram of its nite quotients. A pronite group is a topological group in a natural way: the nite index normal subgroups form a fundamental system of neighbourhoods of the identity. In particular, open subgroups are of nite index. Let G be a pronite group. Let S be a set with a G-action. The following are equivalent: The G-action is continuous for the discrete topology on S. For every point s 2 S, the stabiliser of s is an open subgroup of G. The set S equals S H S H, where H runs over the open normal subgroups of G, and S H denotes the H-invariants of S. The category of G-sets satisfying the above equivalent condition is a Grothendieck topos, denoted G-set. 1

2 The topos Sh(Spec(k) et ) Let k be a eld. Fix a separable closure k=k. Write G for the absolute Galois group, which is by denition the pronite group lim l=k Gal(l=k), where l=k runs over the nite Galois extensions contained in k. Let X=k be an etale k-scheme. Observe that G a on X( k). For every open normal subgroup H G, we have X( k) H = X( k H ). Consequently X( k) = S H X( k) H, and therefore the G-action is continuous. Theorem. The functor Spec(k) et! G-set X 7! X( k) is an equivalence of categories, and the etale topology on the left corresponds to the canonical topology on the right. Proof. We construct a quasi-inverse. Let S be a continuous G-set. Let s 2 S be a point. Write H for the stabiliser of s. It is a open subgroup. By elementary group theory, we have an isomorphism of G-sets G=H! Gs. This shows that S is the disjoint union of nite orbits. We map Gs to Spec( k H ), and since Spec(k) et has arbitrary disjoint unions, this dene a functor G-set! Spec(k) et. It is left as an exercise to show that these functors are actually quasi-inverse. Since we are working with the small etale site Spec(k) et, every morphism in Spec(k) et is etale. It is then immediate that the etale topology coincides with the canonical topology. qed Corollaries. 1. Every sheaf on Spec(k) et is representable. 2. Global sections on Spec(k) et correspond with G-invariants on G-set. 3. Abelian etale sheafs on Spec(k) et correspond to etale commutative k-group schemes, and correspond to abelian G-modules. F1f 2

3 Group cohomology Injective obje Let G be a discrete group. The category G-mod consists of commutative group obje in G-set. For an object A we denote with A G the G-invariant elements of A. Observe that G-mod is equivalent to Z[G]-mod. Let H! G be a group homomorphism. Then there is an adjunction Hom G (A; Ind G H (B)) Hom H(A; B) where Ind G H (B) is the G-module Map(G; B)H. In the category of abelian groups, the injective obje are the divisible groups (abelian groups such that multiplication by n is surjective for all n 2 Z >0 ). It follows from the adjunction that, if A is a divisible group, then Ind G (A) is an injective object in G-mod. In particular, 1 this gives a method to construct injective resolutions in G-mod from injective resolutions in Ab. Computation with cochains For r 2 Z 0, let P r be the free abelian group generated by G r +1. Naturally P r comes with a G-action, by coordinatewise multiplication. Dene boundary operators d r : P r! P rx1, by d r (g 0 ; : : : ; g r ) = rx i=0 (x1) i (g 0 ; : : : ; ^g i ; : : : ; g r ); where ^ denotes the usual `ommision'. One may check that P is a complex. If we denote with " : P 0! Z the map that sends every basis element to 1, we can prove: " Lemma. The complex P! Z! 0 is exact. Proof. Dene r : P r! P r +1 by r (g 0 ; : : : ; g r ) = (e; g 0 ; : : : ; g r ): By construction d r +1 r + rx1 d r = 0. If d r (x) = 0, this implies x = d r +1 ( r (x)). qed Corollary. For every G-module A, we have H n (G; A)æH n (Hom G (P ; A)): 3

4 Observe that Hom G (P r ; A) consists of maps : G r +1! A satisfying (gg 0 ; gg 1 ; : : : ; gg r ) = g(g 0 ; g 1 ; : : : ; g r ): Consequently, to know the value of (g 0 ; g 1 ; : : : ; g r ), it is enough to know the value of (g 1g 0 0; g x1g 0 1; : : : ; g x1g 0 r). Therefore, it is enough to know the value of on tuples of the form (1; g 0; g 0g 0 ; : : : ; g 0g 0 y yg r ). In this way, we obtain a map G r +1! G r (g 0 ; g 1 ; : : : ; g r ) 7! (g 0 1 ; g 0 2 ; : : : ; g 0 r ) g 0 i = g x1 ix1 g i; and this map induces an identication of Hom G (P r ; A) with Map(G r ; A). If we write C r (G; A) for Map(G r ; A), we nd that the induced boundary maps are given by where d r : C r (G; A)! C r +1 (G; A) 7! d r ; d r (g 1 ; g 2 ; : : : ; g r +1 ) = g 1 (g 2 ; : : : ; g r +1 ) + rx i=1 (x1) i (g 1 ; : : : ; g i g i+1 ; : : : ; g r ) + (x1) r +1 (g 1 ; : : : ; g r ): Writing Z r (G; A) for ker(d r ), and B r (G; A) for im(d rx1, we nd H r (G; A)æZ r (G; A)=B r (G; A): This allows for pretty explicit descriptions of H n (G; A) for small n. We give the descriptions for n = 1. The elements of Z 1 (G; A) are called crossed homomorphisms. They are maps : G! A satisfying g(h) x (gh) + (g) = 0: On the other hand, elements of B 1 (G; A) are called principal homomorphisms. They are the maps a for each a 2 A, with a (g) = ga x a: We will now put this description to use. 4

5 Hilbert's theorem 90 If l=k is a nite Galois extension, and G its Galois group, then l { is a G-module. The following theorem by Hilbert goes by the name Hilbert's theorem 90. We have H 1 (G; l { ) = 0. Proof. For this proof, we need the following fact, known as Dedekind's theorem on the independence of characters : Let l be a eld, and G a group. Every nite set of homomorphisms i : G! L { is linearly independent over l. In other words, if P i c i i (g) = 0 for all g 2 G, then c i = 0 for all i. With this fact we can proceed with the proof of Hilbert's theorem 90. Let : G! l { be a crossed homomorphism: (gh) = g(h) y (g); for all g; h 2 G: For all a 2 l {, write b a = X g2g (g) y ga: Apply the above fact to the nite set of homomorphisms (gy): l {! l {. Since (g) 6= 0 for all g 2 G, the fact shows that P g2g (g)g is not the zero-map. We conclude that there is an a such that b a is non-zero. Fix such an a, and write b = b a. For all h 2 G we have X X hb = h(g) y hga = (hg)(h) x1 hga = (h) x1 b: g2g g2g But this means that (h) = b=hb = hb x1 =b x1, which is to say that is principal. qed Galois cohomology For pronite groups, all the above remains true, except that we need to ask most of the maps to be continuous. In particular, we replace C r (G; A), Z r (G; A), and B r (G; A) by their subsets of continuous maps; denoted respectively C r (G; A), Zr (G; A), and Br (G; A). In particular H r (G; (G; A) A)æZr B r (G; A): Observe that there is a natural map colim H C r (G=H; A H )! C r (G; A); given by the compositions with G! G=H, and A H! A. (The colimit runs over the open normal subgroups of G.) We claim this map is an 5

6 isomorphism. Injectivity is clear. For surjectivity, take a continuous map : G r! A. Then (G r ) is discrete, and compact because G r is compact. Hence (G r ) is nite, and thus contained in M H 0 for some normal open subgroup H 0 G. On the other hand, for every a 2 (G r ), the inverse image x1 (a) is open, and thus contains a translate of Ha, r for some normal open subgroup H a G. Now H T 1 = a2(g H r a is an normal open subgroup of G, and by construction factors via (G=H 1 ) r. Finally, write H = H 0 H 1, so that lifts to a map (G=H) r! A H. This proves surjectivity. Because the system of normal open subgroups of G is ltered, the colimits indexed by this system commute with kernels and cokernels, and therefore with cohomology. We thus obtain: H r (G; A)æcolim HH r (G=H; A H ): Corollary We also have a statement of Hilbert's theorem 90 for the absolute Galois group: If k is a eld, with absolute Galois group G, then H 1 (G; k { ) = 0. 6