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with cohomological descent still induces a quasi-isomorphism, even t h o u g h non-filtered. Gluing it to filtered c...
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with cohomological descent still induces a quasi-isomorphism, even t h o u g h non-filtered. Gluing it to filtered cohomology R~( 9 , Q) via the non-filtered version, one gets 7]-Hodge complexes.
9
l-adic Cohomology
Our next aim is the definition of the ~tale or more precisely of the 1-adic realization of a simplicia] variety. We fix an algebraic closure f~ of our ground field k. Let Gk = Gal(k/k) be the absolute Galois group. Very soon, we will have to assume k finitely generated over ~. In addition, we choose a fixed prime l.
9.1
Galois Modules
D e f i n i t i o n 9.1.1 Let 7]/In[Gk] be the category of 7]/l'~7]-modules with a
continuous Z/l'~-linear Gk-operation i.e. it factorizes over the Galois group of some finite extension of k. Let Zz[Gk] be the category of Tlz-modulcs with a continuous operation of Gk. More precisely, we mean the category of projective systems (An)~e~ where A,~ is an object in 7]/l'~[G~]. Morphisms are systems fn ofZ/l'~[Gk] morphisms which are compatible with the structural morphisms of the projective systems.
Let = | Q. This category has the same objects as but on the groups of morphisms the tensor product with Q is taken. It is called the category of Galois modules.
R e m a r k : The category 7]z[G~] is abelian, where the kernels and cokernels are c o m p u t e d componentwise. It has enough injectives ([38] 1.1).i The operation 9 | Q turns an abelian category into another abelian category, and the functor 2~z[Gk] > Qz[Gk] is exact and maps injectives to injectives. A compatible system of elements of the Am can be seen as an "element" of the object (A~)~e~ in 7]z[Gk]. An object A of Qz[Gk] is zero if and only if there is an m E IN such t h a t m 9 An = 0 for all components AN of A. I m p o r t a n t : this condition is stronger t h a n being torsion object in 7]z[G~]. For objects of Ql[Gk] one should not speak of "elements".
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L-ADIC C O H O M O L O G Y
D e f i n i t i o n 9.1.2 An object A in 77z[Gk] is called constructibte if all A,~ are finite and all transition morphisms are of the form A~+I ) A~+l/l '~ = An. The projective limit (formed in the category of abelian groups) is automatically finitely generated as Zz-module.
A Galois module is called constructible if it is isomorphic in Ql[G~] to an object which is constructible as object of Tl[Gk]. We have recapitulated the definition of (constructible) 1-adic sheaves over Spec(k) (cf. [51] Ch. V w p. 163 ff or [27] I 12.3). The category of constructible Galois modules is an abe]ian subcategory of Qz[G~]. Nonconstructible Galois modules are going to used in our constructions. However, they should only be seen as background, as our real interest is in the constructible ones. P r o p o s i t i o n 9.1.3 The category of constructible Galois modules is abelian. It is equivalent to the category of finite dimensional Qt-vect~ spaces with a continuous Gk-operation. (Vector spaces are equipped with the l-adic topology, Gk with the profinite one.) P r o o f i Consider a morphism of constructible objects in 7]z[Gk]. Its cokernel is also constructible. The kernel is not but by the Artin-Rees lemma it is Artin-Rees equivalent to a constructible subobject ([27] I 12.2). But ArtinRees null systems are killed by a power of l, so the kernel of our morphism is a constructible Galois module. There is an obvious functor from the category of constructible Galois modules into the category of continuous finite dimensional Gk-modules: first we assign to an object in Zz[Gk] its inverse limit and then take the tensor product with Q. The functor is also well defined on morphisms. It is essentially surjective because there is a Gk-invariant 771-lattice in a continuous finite dimensional Q,[Gk]-vector space(J61] c a I 1.1 aem. 1)). By choice of such a lattice we get a constructible object in 77z[Gk]. Any morphism respects (after multiplication with the denominator of the matrix entries) these lattices, hence it is image of a morphism in Qg[Gk]. We still have to check that the functor is faithful. A constructible object in 7]l[Gk] whose image in the category of Qrvector spaces is zero has finite inverse limit. Multiplication with the order of the limit annihilates it in
Q,[Ck].
[]
In the category of constructible Galois modules, it is allowed to do computations with "elements".
9.1
Galois Modules
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D e f i n i t i o n 9.1.4 Let k be finitely generated over Q.
a) A constructible Galois module M is pure of weight n if 1. there is a smooth model V of k of finite type over Spec T][}], such that M also extends to a constructible sheaf J~/] on U; 2. and ~/] is pointwise pure of weight n, i.e. on the stalk of M in every closed point x of U, the Frobenius of the residue field operates with eigenvalues of absolute value g ( x ) ~ (the eigenvalues are algebraic and have this absolute value with respect to all embeddings
of Q~ to r b) A filtration on a construetible Galois module N is called a filtration by weights if the n-th graded part is pure of weight n. c) A constructible Galois module which admits a filtration by weights is called mixed. R e m a r k : In the number field case to be pure of weight n means: for all places p of k outside of a finite exceptional set, we have: 1. p is unramified, and 2. the Frobenius of p operates with eigenvalues of absolute value N ( p ) ~ . The properties of the category of mixed Galois modules are studied in [14] by Deligne, the definition is also his (loc. cit. 1.2.2 and 6.1.1. b)). In [39] 6.8. t h e y are collected. Usually, the term weight filtration is used instead of filtration by weights. However, we want to stick to our definition of weight filtration as an ascending filtration on an object of an abelian category. It is the fundamental result of Deligne [14] t h a t our weight filtration on 1-adic cohomology is indeed a filtration by weights. But this will come later. We want to assemble the most important properties: L e m m a 9.1.5 ( J a n n s e n ) Let M and N be constructible Galois modules with the filtrations by weights.
1. The filtration by weights on M is uniquely determined by the mixed Galois module. 2. A morphism f : M - ~ N of mixed Galois modules is strictly compatible with the filtrations by weights.
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L-ADIC COHOMOLOGY
3. The category of Galois modules equipped with their filtrations by weights is abelian. 4. If O ) M )N ) M' ) 0 is an exact sequence of filtered Galois modules (i.e.the arrows are strict) in which the filtrations on M and M r agree with the filtrations by weights then the same is true for N. P r o o f : The first three properties are [39] 6.8.1 a) and b). Because of strictness we can reduce in 4.) to the graded parts of the weight filtration. It is an elementary computation t h a t the eigenvalues of the Frobenii have the right absolute values. []
D e f i n i t i o n 9.1.6 The 11-object in 7]z[Gk] is the system (Z/ln)~E~ equipped with the trivial operation of the Galois group. The Tate-object 11(1) is the projective system of the roots of unity #l. in k with the natural operation of the Galois group.
By the way, 11 and 11(1) are isomorphic as 2[l-modules. The category of constructible Galois modules is a rigid Tannakian category with a faithful fibre functor into the category of ~z-vector spaces. In the same way as in the singular realization, we consider the derived category of Ol[Gk!filt and in it the subcategory of complexes with strict differentials. The category is derivably strict again, hence we get the implications of 2.1.3. Now we have to restrict to the case of fields k which are finitely generated o v e r (~.
D e f i n i t i o n 9.1.7 Let k be finitely generated over Q. Let D~(Gk) be the full subcategory in D+r(Qz[Gk]~lt) of those complexes whose naive cohomology objects are in the category of constructible Galois modules with a filtration by weights.
P r o p o s i t i o n 9.1.8 Dz(G~) is an R-category with weights.
Proof: By 2.1.4 we get a triangulated category with t-structure. The twist is tensor product with Qz(n). (As the Tate-object is fiat, this operation is well behaved under the change to the derived category.) The other properties are clear. []
9.2
First Step
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R e m a r k : In [26] Ekedahl constructs "derived" categories of 1-adic sheaves in a more general situation. Over a point Spec(k), he obtains a category D~(Tlz[Gk]). Jannsen has also given a definition [40]. At least in the case of a point, he gets the same notion. Our ad-hoc construction gives in principle the same thing again, except that we are working with Qz-sheaves and take care of the weight filtration.
9.2
First Step
From now on, let k be finitely generated over Q. We start to construct the realization functor, as before at first with values in a category of complexes. We consider @tale cohomology of constant sheaves. Let ja- be an @tale sheaf of abelian groups on Spec(k). s~c9v is denoted ~'x for short (Sx the structural morphism of the variety X). By adjunction a morphism of varieties f : X ) X' induces a morphism of @tale sheaves .T'x, ~ f..T x. Let Gdx( 9 ) be the functor which assigns to a sheaf of abelian groups on the @tale site of X its Godement resolution. Now we continue as in the definition of singular cohomology. D e f i n i t i o n 9.2.1 Let[~et(. , J ' ) : Y0 ) C+(T7[Gk]filt) (whereT/[Gk] stands for the category of abelian groups with continuous Gk-operation) be the following functor: to an object (U, X ) in ~o, the complex
Re,((U, X),.T) = s x , G d x j , Gdu(.Tu) is assigned. It is equipped with the ddcalage of the filtration induced by the canonical filtration T