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´Etale Cohomology I I H´Et(X,F ) := H (X´Et,F )

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´Etale Cohomology I I H´Et(X,F ) := H (X´Et,F ) Etale´ cohomology Prof. Dr. Uwe Jannsen Summer Term 2015 Inhaltsverzeichnis 1 Introduction 1 2 Grothendieck topologies/Sites 2 3 Constructions for presheaves and sheaves 4 4 The abelian categories of sheaves and presheaves 14 4.A Representable functors, limits, and colimits 17 4.B Filtered categories 30 5 Cohomology on sites 32 6 Spectral sequences 37 7 The ´etalesite 49 8 The ´etalesite of a field 57 9 Henselian rings 62 10 Examples of ´etalesheaves 72 11 The decomposition theorem 80 12 Cechˇ cohomology 86 13 Comparison of sites 94 14 Descent theory and the multiplicative group 98 15 Schemes of dimension 1 105 1 Introduction In mathematics, one often looks for invariants which characterize or classify the regarded objects. Often such invariants are given by cohomology groups. This is a long standing approach in topology, where one considers singular cohomology groups Hi(X; Q) of a topological space X, which are defined by explicit `cycles' and `boun- daries'. These suffice to determine the genus g of a (compact) Riemann surface: If X looks topologically like a sphere with g handles: g=1 g=2 1 then dimQ H (X; Q) = 2g. These cohomology groups can also be obtained as sheaf cohomo- logy (of a constant sheaf). Riemann surfaces can also be regarded as complex algebraic curves, i.e., as algebraic curves over a field C of complex numbers. For any algebraic varieties X over any field k (or any scheme) one can consider sheaf cohomology with respect to the Zariski topology. This is useful for coherent sheaves, for example for the Grothendieck-Serre duality and the Riemann-Roch theorem. However, the Zariski cohomology of an algebraic variety X over C does not give the singular cohomology of a topological space X(C); this is due to the fact that this topology is much finer than the Zariski topology. Furthermore one wants to obtain an analogous topology for varieties over any field k. For fields with positive characteristic, Serre showed that there exists no cohomology theory H∗(−; Q), such that H1(X; Q) has the dimension 2g for a smooth projective curve of genus g. But Weil had postulated such a theory to show the Weil conjectures for varieties over finite fields by a fixpoint formula, as it is known in topology. The solution was found by Grothendieck, together with M. Artin, by creating the ´etale i cohomology. For any prime ` 6= char(k) this provides cohomology groups H (X; Q`) that have the properties postulated by Weil. With these, Deligne eventually proved the Weil conjectures. 1 2 Grothendieck topologies/Sites Grothendieck's approach for the ´etalecohomology (and since then for many other theories) was to leave the setting of topological spaces. He noticed that one only needs the notion of `coverings' with certain properties, to define sheaves and their cohomologies, by replacing at the same time `open set' by `object in a category'. Definition 2.1 Let X be a category and C another category. A presheaf on X with values in C is a contravariant functor P : X!C : Morphisms of presheaves are morphisms of functors. (Here we ignore {actually non-trivial { set theory problems by assuming that the category X is small). If C is the category Ab of abelian groups (resp. the category Rg of rings, resp. ...), then one speaks of presheaves of abelian groups [for short: abelian presheaves] (resp. of rings, resp. ...). Example 2.2 Let X be a topological space. Then one can assign to X the following category X: Objects are open sets U ⊆ X. Morphisms are the inclusions V ⊆ U. Then one can see that a presheaf in Grothendieck's sense is just a classical presheaf: Because of the contravariant functoriality, one has an arrow P (U) ! P (V ) for every inclusion V ⊆ U. The properties of a functor provide the properties of presheaves for these `restrictions' resU;V . Definition 2.3 Let X be a category. 'i (a): A Grothendieck topology on X consists of a set T of families (Ui ! U)i2I of mor- phisms in X , called coverings of T , such that the following properties hold: (T1) If (Ui ! U)i2I in T and V ! U is a morphism in X , then all fibre products Ui ×U V exist, and (Ui ×U V ! V )i2I is in T . (T2) If (Ui ! U)i2I is in T and (Vij ! Ui)j2Ji is in T for all i 2 I, then the family (Vij ! U)i;j obtained by the compositions Vij ! Ui ! U is in T . ' (T3) If ' : U 0 ! U is an isomorphism, then (U 0 ! U) is in T . (b) A site is a pair S = (X ; T ) with a category X and a Grothendieck topology T on X . One denotes the underlying category X also by Cat(S) and the topology also by Cov(S), thus S = (Cat(S); Cov(S)). Sometimes (X ; T ) is called a Grothendieck topology as well. Example 2.4 If one takes the usual coverings (Ui)i2I of open sets U ⊆ X in example 2.2, then the corresponding families (Ui ,! U)i2I form a Grothendieck topology on X. Note: The fibre product of open sets U ⊆ X; V ⊆ X is the intersection U \ V . Definition 2.5 Let S = (X ; T ) be a site, and let C be a category with products (e.g., the category of sets or of abelian groups). A presheaf F : X!C 2 is called a sheaf (with respect to T ), if for every covering (Ui ! U)i2I in T the diagram α Q α1 Q F (U) ! F (Ui) ⇒ F (Ui ×U Uj) i α2 i;j is exact, where the arrow α1 on the right side is induced by the first projections Ui ×U Uj ! Ui and the arrow α2 is induced by the second projection Ui ×U Uj ! Uj (This means that α is the difference kernel of α1 and α2, see appendix 4.A below). Morphisms of sheaves are morphisms of the underlying presheaves. Remark 2.6 Let C be the category of sets. If, for s 2 F (U), we denote the component of Q α(s) in F (Ui) by sjUi and for (si) 2 F (Ui), we denote the images of si and sj in F (Ui ×U Uj) i by sijUi×U Uj and sjjUi×U Uj respectively, then we literally obtain the same conditions as for the usual sheaves on topological spaces, except that we replace Ui \ Uj with Ui ×U Uj: The conditions are: (i) If s; t 2 F(U) and sjUi = tjUi for all i, then s = t. Q (ii) If (si)i2I 2 F(Ui) with sijUi×U Uj = sjjUi×U Uj for all i; j 2 I, then there is an s 2 F(U) i with sjUi = si for all i 2 I. Definition 2.7 (a) A morphism f :(X 0; T 0) ! (X ; T ) of sites is a (covariant) functor f 0 : X!X 0 (!) which has the following properties: 0 'i 0 f ('i) 0 0 (S1) If (Ui −! U) is in T , then (f (Ui) −! f (U)) is in T . (S2) If (Ui ! U) is in T and V ! U is a morphism in T , then the canonical morphism 0 0 0 f (Ui ×U V ) ! f (Ui) ×f 0(U) f (V ) is an isomorphism for all i. Example 2.8 If f : X0 ! X is a continuos map between of topological spaces, we obtain a morphism f : S(X0) !S(X) of the associated sites (Example 2.4) by f −1 : X ! X0 U 7! f −1(U) : 3 3 Constructions for presheaves and sheaves For a category X let P r(X ) be the category of abelian presheaves on X . Definition 3.1 (Push-forward) Let f :(X 0; T 0) ! (X ; T ) be a morphism of sites and let 0 0 0 0 P : X ! Ab be an abelian presheaf. Then the direct image (or push-forward) fP P of P is defined as the presheaf 0 0 0 0 0 f 0 P fP P = P f : X !X ! Ab : 0 0 0 0 0 0 0 Explicitly we have (fP P )(U) = P (f (U)) for U in X and fP (') = P (f (')) : P (f (U2)) ! 0 0 0 0 P (f (U1)) for ' : U1 ! U2 in X ). For a morphism : P1 ! P2 of abelian presheaves on X one obtains a morphism 0 0 (3:1:1) fP : fP P1 ! fP P2 as follows: For U in X define 0 0 (fP )U :(fP P1)(U) ! (fP P2)(U) q q q 0 0 0 0 f0(U) : P1(f (U)) ! P2(f (U)) : One can see easily that this produces a morphism of presheaves (3.1.1) and that one obtains a functor 0 fP : P r(X ) ! P r(X ) 0 0 P 7! fP P 7! fP : Proposition 3.2 The functor 0 fP : P r(X ) ! P r(X ) has a left adjoint f P : P r(X ) ! P r(X 0) : For presheaves P 2 P r(X ) and P 0 2 P r(X 0) we thus have isomorphisms P 0 ∼ 0 (3:2:1) HomX 0 (f P; P ) = HomX (P; fP P ) ; functorially in P and P 0. For a presheaf P on X , f P P is called the inverse image (or pull- back) of P . 0 0 Proof of 3.2: For U in X consider the following category IU 0 : Objects are pairs (U; ), where U is an object in X and 0 0 : U ! f (U) 0 is a morphism in X .
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