In this thesis, we first briefly introduce the history of the Weil-'etale cohomology theory of arithmetic schemes and review some important results established by Lichtenbaum, Flach and Morin. Next, we generalize the Weil-'etale cohomology to S-integers and compute the cohomology for constant slices Z or R. We also define a Weil-'etale cohomology with compact supportHc(YW,−) forY = SpecOF,S where F is a number field and calculated them.
Recall that Lichtenbaum only defined the Weil-´etal cohomology for SpecOF, but you can generalize it to arbitrary open subschemes Y = SpecOF,S in an obvious way. In Chapter 2, we compute the Weil-'etal cohomology of Y with coefficients Z and give a brief summary of Morin's results on Weil-'etal cohomology. Under this assumption, in Section 3.1, we show that Weil-´etal cohomology groups with compact support for integers S meZ or recoefficients are as follows.
Therefore, it may be possible to define a certain Weil-´etale cohomology for an arbitrary arithmetic scheme by generalizing RGm. Therefore, we see that the complex RΓ(U´et, RGm) generalizes Tate sequences and its Z-dual recovers the truncated Weil-´etals of the cohomology group of S-integers.
The Weil Groups for Number Fields and Local Fields
Weil Groups
Let K/F be a finite Galois extension, and S a finite set of places of K containing all the infinities and those branched into K/F, we define WK/F,S as the extension CK,S : = CK /UK,S by Gal(K/F) representing the canonical class αK/F,S of the group.
Weil Maps
It is not difficult to show that the image WFv in WK/F,S is isomorphic to WKw/Fv (or rather, this implies the canonical mapping πv : WFv → Wk(v) factors through WgFv, where WgFv is the image of WFv in WK/P,S.
The Classifying Topos and Cohomology of a Topological Group
Artin-Verdier Topology
The Definition
In the proof of the above property, we actually proved that F ∈ Sh(U´et) vanishes if and only if the stems Fx = 0 for all x ∈ U. Furthermore, to verify the accuracy of complexes of ´etals it is enough to check it on stems. By the construction of mapping cylinder, the section function Γ(−,F) is the sheaf that sends an ´etal W over U to the Abelian group.
The Norm Maps
Let π : V → U be the same as before, and assume that F is completely imaginary, then N(j∗F) is a quasi-isomorphism for all Z-constructible sheafs F on V0.
The ´ Etale Cohomology with Compact Support
Let K/F be a finite Galois expansion and S a finite set of non-trivial valuations of F, containing all that branch into K/F. We consider Xv to be a WFv space via πv, and Xv0 to be a WFv space via Θv). Let pK,S be the morphism BWF →BWK/F,S and A be an abelian object of BWF associated with a topological abelian WF group A .
We define Hp(YW, A) as Hp(YW,A).e In particular, if A is a topological abelian group on which WF acts trivially, then Aeis is called a constant A-valued sheaf. Recall that WgKv denotes the image of WKv in WK/F,S, and θv :WgKv →WK/F,S and qv :WgKv →Wk(v) are induced continuous mappings. We can show that FK/F,S0/U is equivalent to T^K/F,SU 0, where TK/F,SU 0 is defined similarly to TK/F,S0 by extracting from each object all v-components for in ∈Y − U.
Since U = Y we define the Weil ´etale cohomology with compact support for an open subscheme U with coefficient F as the cohomologies of the mapping cone.
The Weil-´ Etale Sites and the Artin-Verdier ´ Etale Sites
Applying RΓ(Still´,−) to the right triangle above and using the previous theorem, we obtain the desired right triangle. We will compute the cohomology groups Hc∗(YW,Z) and Hc∗(YW,Re) in Section 3.1 and prove that the axioms of Weil-´etal cohomology theory hold for Y in Section 3.2. Let A be an abelian group with trivial action WF and A be a constant sheaf defined by it.
Certainly. 2.5) the Weil-´etal cohomology groups with compact support Hc∗(YW, A) are defined by the cohomologies of the mapping cone. For each constant sheafA, the uptake A[0]→RA induces a morphism RΓ(Y´et, A)−→RΓ(YW, A), and the following diagram shuttles. Thus, taking the truncation function τ≤3 for each term gives rise to the following commutative diagram.
By Theorem 2.1, this is the same as the morphism of complexes of Yet sheaves. JK,S• ) is any fixed injective resolution of A (resp. i∗K,SA) and note that iK,S,∗ is exact and injective preserving. Applying the natural functor id → i∗i∗ to RA → R0A, we obtain the following canonical commutative diagram. When there is a square commutative diagram of complexes in an abelian category A, one can complete it as a semi-commutative 3 by 3 diagram of complexes in which rows and columns are exact triangles in the derived category D by mapping cones take. (A).
Looking at a long exact sequence of cohomologies, we can identify RΓgc(Y´et, R2Z) with Hom(US,Q)[−1]. By the definition of cohomologies with a compact support, we have the following long exact sequence. By direct computation of the long exact sequence defining cohomologies with a compact support, we see that
Let Y ⊃S1 ⊃S0 ⊃Y∞ and F be a compatible system of sheaves on the sites (TK/F,S), then we have an exact triangle. For A=Z, split the long exact sequence induced by the exact triangle in the above corollary into short exact sequences. Therefore, the morphism of long exact sequences induced by the morphism of the 2nd and 3rd sequences in diagram (3.2) looks like
Verification of the Axioms of Weil-´ Etale Cohomology
- Axiom (a)
- Axiom (b)
- Axiom (d)
- Axiom (c) and (e)
Thus, by the property of exact triangles, the left vertical morphism is also a quasi-isomorphism. Consider the compatible system ((iv,∗i∗vFL,S); (iv,∗i∗vft)), and take mapping cone, we can indeed define a cup product. Note that iv is an embedding and preserves injectives, so we have the following commutative diagram.
Therefore, the middle vertical map of diagram (3.3) is also an isomorphism. b) The rows of (3.3) are induced by spanning the exact sequences with R.
A Canonical Representation of Tate Sequences
We claim that the following diagram commutes for a suitable choice of the left vertical quasi-isomorphism. Since both horizontal maps induce the canonical projection XS⊗ZQ→XS⊗ZQ/Z on H2, the right vertical quasi-isomorphism induces the identity map on H2 and the left vertical one can be induced by any G-automorphism of XS ⊗ZQ, so there is a suitable choice of the quasi-isomorphism so that the above diagram commutes for H2 groups. On the other hand, applying RHomZ(−,Z) to exact triangle (3.1), we have another exact triangle.
Taking the long exact sequence induced by the upper exact triangle, we see that H2(β) = i◦p where p : UbS → UbS/US is the canonical projection and i ∈ Aut(UbS/US). We also assume that K is a subfield of L such that the extension L/K is Galois and G:= Gal(L/K). Also, when U is small enough for the Tate sequence to exist, then by (4.1) and using Theorem 3.3, one can see that.
One can thus conclude that the complex RΓ(U´et, RGm) generalizes the Tate series and that its Z-dual defines the Weil-´etal cohomology of S-integers without truncations. Moreover, for any finite extension K over L, the transition map H3(u(Kv)et´,Gm) → H3(u(Lv)´et,Gm) is an isomorphism. Note that the cup products are canonical on v and w, and we saw above that the bottom cup product is a canonical projection.
If we follow the diagram, we see that the middle one is also a canonical projection, and so is the top one, since the injection is canonical. As we saw at the very beginning of the proof that ∂U1 and π are canonical projections, ∂U1 must be an identity mapping onto Q/Z with commutativity. Consider the usual long exact sequence on the cohomologies induced by the exact triangle RGm → F•[−1] −→γ Gm[1] → and use the result that ∂i are surjective (Lemma 4.4), which we get.
The Duality Theorem
The upper right and lower right squares of the above diagram are commutative because they are commutative on H0 and H2 respectively (see Lemma 3.2). It is easy to see that the exact triangle associated with the next exact sequence of complexes is in fact isomorphic to the middle row of the above diagram. We conclude that RΓ(U´et,Gm) can be represented by A→B →F(U) and this induced the exact triangle.
Suppose Pic(Spec(OF,S)) = 0 and S is stable under the action of G, then we have the following quasi-isomorphism. Also, one can find corresponding Weil-'etal cohomology axioms in terms of Hp(U'et, RGm). The reason why it is difficult to generalize Lichtenbaum's prototype to higher dimensional arithmetic schemes X is that there are no Weil groups for higher dimensional fields.
However, Theorem 4.3 shows us a probability of generalizing Lichtenbaum's prototype because we do not use Weil groups when defining the complex RGm. A direct thought is that for any n-dimensional arithmetic scheme X, one can define a complex RZ(n) in Ext2X(Z(n),F•(n)), so that the Z-dual of RΓ(Xet, RZ (n)) defines a certain Weil-´etale cohomology theory, where F•(n) is a complex of ´etale slices depending on n. Jensen, Les Foncteurs D´eriv´es de lim←− et leurs Applications en Th´eorie des Modules, Lecture Notes in Math., 254, Springer, Heidelberg.
