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An Abelian Category of Motivic Sheaves

The goal of this paper is to construct a category of motivic"sheaves"on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful exact realization functors to the categoriesof constructible sheaves for the classical and etale topologies. Moreover, there is a tannakian subcategory of motivic local systems with a realization functor into the category of variations of mixed Hodge structures. Conversely, all basic geometric examples of the latter come from this motivic category.


Introduction
The basic homological invariants of a fibration of topological spaces f : X → S, are the local systems R i f * Q. When this is a family of complex algebraic varieties defined over a subfield k of C, there are many related invariants, such as the Gauss-Manin connection, the associated variation of mixed Hodge structure, and the action of the algebraic fundamental group onétale cohomology of the fibres. According to Grothendieck's philosophy, all of these structures should come from the motive of the family. My goal here is to make this idea precise in the following way. Given a field F , and a variety S, as above, I will construct an abelian category M(S; F ) of motivic "sheaves" of F -modules. The above local systems can be promoted to objects in M(S; Q), and the associated structures can be obtained by applying appropriate realizations functors.
Before explaining what I will do, let me say a few words about what I won't. The usual approach to building a category of motives is to start with a category of varieties and algebraic correspondences and modify and complete this in some way. This stays very close to the underlying geometry which is good. On the other hand, it is usually very hard to prove for example that what one gets is (derived from) an abelian category. A more pragmatic approach is to take a system of compatible realizations. This usually has good categorical properties, but is somewhat ad hoc in nature; and in the relative setting, it would be appear that any such approach would be necessarily very technical (e.g. [S3]). Here I want to take a middle path first blazed by Nori while building a category of motives over a field [N2, L2]. The approach appeals to a particular realization at the outset, but is essentially geometric in its character. The building blocks for M(S; F ) are quadruples consisting of a family f : X → S, a closed subvariety Y ⊂ X and indices i ∈ N, w ∈ Z. When Y = ∅, this data represents the motivic version of R i f * F denoted here by h i S (X)(w). Although the parameter w is extraneous at this stage, it is needed to keep track of Tate twists, which are nontrivial in the Hodge andétale realizations. For nonempty Y , the associated motive h i S (X, Y )(w) roughly corresponds to the fiberwise cohomology of the pair. In essence, M(S; F ) is set up as the universal theory for which: 1. M(S; F ) is an F -linear abelian category with a faithful exact functor R B to the category of sheaves of F -modules on S with its classical topology.

2.
A morphism X ′ → X over S, taking Y ′ to Y would give rise to a morphism As with most universal objects, the construction is rather formal. A category PM(S, Σ; F ) of premotivic sheaves constructible with respect to a fixed stratification Σ is constructed explicitly as the category of comodules over a coalgebra built from the above quadruples. PM(S; F ) is then given as the direct limit of these categories. This will satisfy the first 4 conditions, and the category M(S; F ) is obtained by forcing the last condition as well. When S = Spec k, this should almost certainly coincide with Nori's category of cohomological motives over k tensored with F . Here are the precise properties: to the heart of the classical t-structure of the derived category mixed Hodge modules (see appendix C). 5. There exist motives in M(S; F ) corresponding to R i f * F (n) under realization.
6. There are inverse images compatible with realizations.
7. There are higher direct images (under some conditions) compatible with realizations.
8. There are tensor products on M tls (S; F ) compatible with realizations, making this into a Tannakian category when dim S ≤ 1.
9. The subcategory M pure (S, Q) ⊂ M tls (S, Q) generated by smooth projective families is a semisimple Tannakian category when dim S ≤ 1 The construction of direct images is technically the most difficult part, and for the moment I've had to settle for a weaker statement that I was hoping for. There are a number of other loose ends as well. I hope to fix all of this up in the next version. Also to simply the task of writing I have made two restrictions which are probably unnecessary. I work with quasi-projective varieties, and with coefficients in a field.
I would like to thank Madhav Nori for giving me his permission to include some of his beautiful constructions (of course, I take responsibility for errors). Also thanks to M. de Cataldo, F. Déglise, M. Levine, D. Patel and J. Schürmann for some useful conversations. This research was started at the Max Planck Institute in the fall of 2007 and continued at the Tata Institute in the winter of 2008; I am grateful to these institute as well as the NSF for their support. Notation: Since the notation will tend to get rather heavy, I will routinely suppress subscripts, superscripts and others symbols whenever they can be understood from context. Given a ring R, let R-Mod (respectively R-mod) stand for the category of (finitely presented) left R-modules. Fix a field k embeddable into C and another field F . For most of the paper, I will work with a fixed embedding ι : k ֒→ C. A k-variety is simply a reduced separated k-scheme of finite type. Let V ar k be the category of these. Given a k-variety X, I will denote the analytic space (X × ι Spec C) an by X ι,an or X an or sometimes just X. I will usually write H i (X; F ) for H i (X ι,an ; F ). Given a map f : X → S of spaces and a sheaf F on X, I will often denote the higher direct image R i f * F by H i S (X, F ). Since this will never be used to denote cohomology with support in this paper, there should be no danger of confusion. I likewise denote R i f ! F by H i c,S (X, F ).

Representations of graphs 2.1 Endomorphism coalgebras
In the next couple of sections, we set up the basic foundation for the rest of the paper. Let F be a field. Suppose we are given an F -linear abelian category A with an exact faithful embedding H into the category of finite dimensional vector spaces F -mod. Then the ring End(H) will act naturally on H(A) for any A ∈ ObA. This suggests that one might be able to reconstruct A as the category of modules over this ring. However, this does not quite work.
The right thing to consider is the category of comodules over the dual object End ∨ (H) whose construction we learned from [JS]. To clarify, this is really the "predual" in the sense that Nevertheless, most of the statements become easier to follow if one formulates them for End(H) and "dualizes". A precise formulation is given below. Note that we can express End ∨ (H) or any coalgebra as a directed union of finite dimensional subcoalgebras ∪E i . Thus the correct dual object to End ∨ (H) is not End(H) but the pro-algebra " lim ← − "E * i . Moreover, A can described as 2colimit of the categories of E i -modules. We will find this viewpoint convenient later on, but for the moment, it seems simpler to work with the coalgebra.
Given pair of functors G, H : where the map takes the collection (η M H) into an algebra as noted above. Following [JS], it is convenient to introduce a smaller predual" object, which means that Hom ∨ (G, H) * = Hom(G, H). Let F -Lin be the collection of F -linear abelian categories with finite dimensional Hom's. Suppose that we now have a pair of functors G, H : C → D, with D ∈ F -Lin. Define Hom ∨ (G, H) to be the cokernel of where the map S is defined so that dual to the identity, is easily seen to factor through End ∨ (G) := Hom ∨ (G, G) and this defines the counit dual to composition •. More precisely, • is given by product of compositons Then • ∨ is given by the sum of the dual maps c * dual to the operation ⋄ defined in the appendix. The operation ⋄ is a product of maps To simplify arguments with these operations, we use the following duality principle: Lemma 2.1.1. Suppose we are given an identity in +, •, ⋄, 1 G , which amounts to the commutivity of a finite diagram with arrows labelled by these operations. Then the dual identity, obtained by reversing arrows and relabelling by +, • ∨ , ⋄ ∨ , 1 ∨ G , also holds. Proof. Suppose we have a finite diagram with vertices given as finite tensor products of spaces Hom ∨ (−, −), and edges labelled by +, • ∨ , ⋄ ∨ , 1 ∨ G . Then commutivity can be established by chasing elements. Given an element of one of the vertices, we can find a subdiagram of finite dimensional vector spaces which contains it. Duality for finite dimensional vector spaces implies that the commutivity of the subdiagram would then follow from commutivity of the dual diagram.

The map p given by
is naturally an End(H(M ))-module, and hence by transpose an End(H(M )) * -comodule. Via the map End(H(M )) * → End ∨ (H), M becomes a End ∨ (H)-comodule, which we usually denote by h (M ). This is a functor ∆ → End(H)-comod. The structure of a general comodule is clarified by the following.
Proof. Set E ∨ (D) = End ∨ (H| D ) for any subgraph. When D is finite, E ∨ (D) has a presentation by a finite sum of comodules of the form Hom(H(P ), H(N )) ∼ = H(N ) dim H(P ) by definition. This proves the lemma for V = E ∨ (D). In general, the matrix coefficients of the E ∨ (∆) coaction on V lie in some E ∨ (D) with D finite. Thus V has a presentation by a finite sum of copies of E ∨ (D).
Remark 2.2.6. There is a dual description of End ∨ (H)-comod which is closer to what Nori originally used [N2]. As in the previous argument, we can express E-comod = ∪E ∨ (D) as D ⊂ ∆ runs over finite subgraphs. Therefore as explained in appendix A, we have equivalences We can use this to deduce a version of Nori's Tannakian theorem. (The original statement, which is stronger, can be found in [Br,thm 1] or [N2].) Corollary 2.2.8 (Nori). Suppose that A is an F -linear abelian category equipped with a faithful exact functor U : It is convenient to prove a slightly stronger statement, where F -mod is replaced by, for example, the category of finite dimensional modules over an Falgebra.
Corollary 2.2.9. Let R be an F -linear abelian category with a faithful exact functor ρ : R → F -mod. Suppose that H : ∆ → F -mod factors as H 1 • ρ (up to natural equivalence). Suppose that A is an F -linear abelian category equipped with a faithful exact functor U : Proof. We obtain a commutative diagram of coalgebras Thus we get a functor between the categories of finite dimensional comodules The equivalences of categories, indicated by ∼, follow from above.
The case of particular interest to us in the above corollary is the category R = (F -mod) n of finite dimensional vector spaces admitting gradings of the This can be identified with the category of finite modules over the ring F n . A natural example arises as follows. Given functors H i : ∆ i → F -mod, we can define a new functor H 1 ×H 2 ×. . . H n on the cartesian product ∆ 1 × ∆ 2 × . . . ∆ n in the category of graphs, to (F -mod) n by We have an induced functor We will need a criterion for when this is an equivalence. It usually isn't.  2 is the diagonal embedding. Lemma 2.2.11. Suppose that each ∆ i has an initial object ∅ satisfying H i (∅) = 0. Then (4) is an equivalence.

Enriched model
Let H : ∆ → F -mod be a functor on a graph as above. Although functors on End ∨ (H)-comod can be constructed with the help of corollary 2.2.9, it is sometimes difficult to apply. It will be convenient to give an alternative description (up to equivalence) of End ∨ (H)-comod which allows us to incorporate extra structure. Let p i be the projections. Suppose that H # : ∆ → F -mod 2 be functor such that H = p 1 • H # . Then we wish to describe the relationship between End ∨ (H # )-comod and End ∨ (H)-comod. Given an F -linear abelian category C, an ideal I is a collection of subspaces I(c 1 , c 2 ) ⊆ Hom(c 1 , c 2 ) such that Given an ideal I, C/I is the category with the same objects as C and is an ideal. Note, however, that the quotient C/ ker G should not be confused with the quotient of C by the thick subcategory generated by {c ∈ ObC | G(c) = 0}. Then D is equivalent to C/ ker G.
Proof. To see that p 1 satisfies these conditions, identify it with the change of scalars functor

Products
We need to incorporate tensor products into our story. The category of functors from graphs to F -mod forms a category with tensor product given as follows. Let H : ∆ → F -mod and H ′ : ∆ ′ → F -mod be two such functors. Then H ⊗ H ′ : ∆ × ∆ ′ → F -mod is given by (M, N ) (N ). The one point graph { * } with * → F gives the unit making this into a tensor category, where for our purposes a tensor category over F is an F -linear additive category with a bilinear symmetric monoidal structure. We have [JS,§8,prop 1]. This yields a product When H = H ′ is equipped with a symmetric associative pairing H ⊗H → H and a unit * ∈ Ob∆, H( * ) = F . Then End(H) becomes a commutative bialgebra. Thus End(H)-comod becomes a tensor category with a tensor preserving functor to F -mod given by the forgetful functor. With minor modifications to the proof of corollary 2.2.8, we have Corollary 2.4.1. Suppose that H has a product as above. If in the hypothesis of corollary 2.2.9, R = F -mod, A is an F -linear abelian tensor category, and the functors ρ, U, G are product preserving. ThenG is also product preserving.
Recall [De3, sect 2] [L1, chap IV, sect 1] that a dual of an object M in a tensor category, with unit 1, is an object M ∨ equipped with morphisms δ : yield the identities. Alternatively, M ∨ is characterized by the natural isomorphisms In particular, the dual is unique up to isomorphism if it exists. A map f : M → N yields a dual or transpose map f ∨ : N ∨ → M ∨ if M, N both possess duals. Lemma 2.4.2. Given an exact sequence and (D4) is similar.
A neutral Tannakian category over F is an abelian tensor category over F , with a faithful exact tensor preserving functor to F -mod, such that every object possesses a dual. Such a category can be realized as the category of comodules over a commutative Hopf algebra.
Proposition 2.4.3. Suppose that H : ∆ → F -mod is equipped with a symmetric associative product as above. Assume that for every object M ∈ Ob∆, h(M ) has a dual. Then End ∨ (H)-comod is neutral Tannakian.
Proof. The proposition follows from lemmas 2.2.5 and 2.4.2 3 Motivic sheaves

Cohomology of pairs
Let S be a C-variety. Let V ar 2 S be the category whose objects are pairs ( Equivalently, this is the sheaf associated to Definition 3.1.1. We say that H i S (X, Y ; F ) commutes with base change if the stalk is given by for every s ∈ S.
Definition 3.1.2. More generally, given a sheaf F ∈ Sh(X), we say that H i S (X, F ) commutes with base change if the stalk is given by for every s ∈ S.
The first condition holds for all i when 1. X → S is proper by the proper base change theorem [I, thm 6.2].
2. or (X, Y ) → S is a topologically locally trivial fibre bundle.
We can combine these two cases: where h is proper and there exists an open cover {U j } of S an such that is homemorphic to a product. Then H i S (X, Y ; F ) commutes with base change. Proof. We can assume that S = U j . Then this reduces the Künneth formula and the proper base change theorem.
Proof. The hypothesis implies that there is an isomorphism on stalks.

Premotivic sheaves
The category PM(S) of premotivic sheaves is constructed as a direct limit of categories PM(S, Σ). Each PM(S, Σ) is obtained by applying the previous construction to an appropriate graph ∆(S, Σ) and functor H Σ given below.
Let S ∈ ObV ar k be connected. Then we construct a graph Γ(S) as follows. The objects (i.e. vertices) are quadruples (X → S, Y, i, w) consisting of 1. a quasi-projective morphism X → S i.e. an open immersion followed by a projective morphism, 2. a closed subvariety Y , 3. a natural number i ∈ N and an integer w.
We require that H i S (X, Y ; F ) commutes with base change. The set of morphisms (edges) is the union of the three following sets: for every chain Z ⊆ Y ⊆ X of closed sets.
Type III: Twisted projection morphisms For arbitrary S ∈ V ar k , set Γ(S) = Γ(S i ), where S i are the connected components. Thus the parameters i and w are locally constant.
By a good stratification or simply stratification of S, we mean a finite partition Σ into connected locally closed sets such that Σ contains the closure of every element. Given a stratification Σ, let ∆(S, Σ) ⊂ Γ(S) be the full subgraph consisting of objects such that H i S (X, Y ; F ) is constructible with respect to the stratification Σ, i.e. it has finite dimensional stalks and its restriction to each stratum σ ∈ Σ is a locally constant. We say that ( Let s = (s σ ) denote a collection of base points, one for each σ ∈ Σ. Let |Σ| be the cardinality of Σ. Define to be the product of stalks. We usually suppress the symbols Σ, s, F . We want to extend H = H Σ,s,F to a functor ∆(S, Σ) op → F -mod. We do this case by case.
gives rise to the natural homorphism for each U ⊂ S. Since this is clearly a morphism of presheaves, it induces a morphism of sheaves by taking the product of this sheaf map over stalks. We give a second description which is a bit more complicated, although better for comparing to theétale case. We have a commutative diagram where the triangles are distinguished, and the solid vertical arrows are the adjunction homomorphisms. Thus we get the dotted arrow above. From which we obtain which is easily seen to coincide with the previous map.
given by exterior product with the fundamental cycle of (P 1 , {0}). This gives rise to Thus we can apply the construction from the previous section to obtain: Definition 3.2.1. The category PM(S, Σ, s; F ) of Σ-constructible premotivic sheaves of F -modules on S is the category of finite dimensional right comodules over End ∨ (H Σ,s ) When S is equipped with the trivial stratification {S}, we sometimes write PM ls (S, s; F ) and call these premotivic local systems. For any finite commutative F -algebra R, let PM(S, Σ, s; F ) ⊗ F R denote the category with finitely generated right comodules over End ∨ (H) ⊗ F R.

Realizations
By definition this carries a forgetful functor U : for a finite subgraph D. Thus V can be regarded as an End ∨ (H| D )-comodule, or equivalently an End(H| D )-module. In fact, we can describe PM(S, Σ, s) as the direct limit of the categories of finite dimensional End(H| D )-modules, as D ⊂ ∆(S) varies over finite subgraphs (cf [Br]). This dual description was employed by Nori in his work, and it would appear that PM(Spec k, Spec C) is just Nori's category of cohomological motives tensored with F . We write this as PM(k; F ) or simply PM(k) from now on.
if w = 0 (we will see shortly that this independent of Σ and s in a suitable sense). When S = Spec k, we omit the subscript.
By definition we have In view of the following, we may suppress base points. Proof. Given a homotopy class of paths γ σ in σ joining s σ to t σ , parallel transport along these curves yields an isomorphism of fiber functors H s ∼ = H t .

Remark 3.3.3. This business of choosing base points and then suppressing them is a bit clumsy. A more elegant approach is to simply redefine
where π σ :σ → σ are the universal covers, and then build PM(S, Σ; F ) accordingly. However, the original approach does make certain things more transparent, and will generally be preferred.
We have the following consequences of corollary 2.2.9.
Construction 3.3.4. Let Constr(S ι,an , Σ; F ) denote the category of sheaves of F -modules which are constructible with respect to the stratification Σ. The fibre functor Φ : Constr(S an , Σ) → F -mod given the product of stalks at the base points provides a faithful exact functor. The previous discussion shows Σ) op and that H is a composition of this with Φ. Thus corollary 2.2.9 yields an extension functor extends to a functor ∆(S, Σ) op → PM(S, Σ) satisfying t n t m = t n+m . When composed with the forgetful functor to F -mod, we obtain H. Thus this extends to an endofunctor T n : PM(S, Σ) → PM(S, Σ) satisfying and T n T m = T n+m ; in particular, it is an automorphism. Construction 3.3.6. Let F be finite or Q ℓ . Letk denote the algebraic closure of k. Consider the map where the sheaves are on theétale topology,X = X × Spec k Speck jXȲ represents the inclusion of the complement as above, and w measures the Tate twist. This is easily seen to be a functor by modifying the above discussion. Thanks to the comparison theorem betweenétale and classical cohomology (appendix B).
Thus we can obtain an extension which is theétale realization which can be composed with above functor Φ to obtain a fibre functor. Consider the map where the operations are in derived categories of mixed Hodge modules, and c H i is cohomology with respect to the classical t-structure. When composed with the forgetful functor to constructible sheaves, we obtain H i S (X, Y ). Thus we obtain a Hodge realization functor A special given in section 5.1 can be made more explicit.
We fixed an embedding ι : k ֒→ C at the outset. Let write PM(S; F ) ι for the resulting category. We now show that the category PM(S; F ) is independent of this. Proof. It suffices to show that PM(S, Σ, F ) ι and PM(S, Σ, F ) µ are equivalent for every stratificaion. We note that PM(S, Σ, ; F ) = PM(S, Σ; F 0 ) ⊗ F0 F for any subfield. Thus it suffices to assume that F is the prime field. Suppose that F = Z/pZ or Q. Then, we can see this immediately by the comparison theorem This description is independent of the embedding. Therefore PM(S, Σ, F ) ι and PM(S, Σ, F ) µ are equivalent. The remaining case F = Q follows Nori's argument [N2] quite closely. Define the category T of triples (A, where morphisms are compatible pairs of linear maps. Write H ι and H µ for the functors corresponding to the embeddings. If p denote the first projection, then p is easily seen to be fully faithful and essentially surjective. Therefore it is an equivalence. So there is functor q : Q-mod → T and natural isomorphisms γ : q • p ∼ = 1 T and η : p • q ∼ = 1 Q-mod . We get a functor H T : ∆(S, Σ) → T by taking these as the first and second component. For the third, we use the composition of the comparison maps . We claim that this gives an isomorphism. Here we use the duality principle given in lemma 2.1.1. The dual of p is given by p * (f ) = 1 p ⋄ f . The map is injective as it has a left inverse given by The map p * is also surjective, because Lemma 3.4.1. Let S = S 1 ∪ S 2 ∪ . . . S n be a decomposition into connected components. Choose stratifications Σ i of S i and let Σ = Σ i . Then

Base change
Proof. This is an immediate consequence of lemma 2.2.11.
It follows that there are projections from the lemma that there are inclusions A morphism of (pointed) stratified varieties is a morphism of varieties such that a nonempty preimage of any stratum is a union of strata (and base points go to base points). If the underlying morphism of varieties is the identity, we say that the first stratification refines the second. Suppose that f : (T, Λ, t) → (S, Σ, s) is a morphism of pointed stratified varieties.
yields an extension, which is the base change functor f * : which is f on T and identity on s ′ , satisfies (*). We now define f * as the composite where p is the projection.
We can always extend a morphism of stratified varieties to a morphism of pointed stratified varieties, and this way obtain base change functor f * : PM(S, Σ) → PM(T, Λ). Alternately, we could define this directly in the spirit of remark 3.3.3 without recourse to base points. We mention two important special cases of this construction: 1. The construction applies to refinement, leading to faithful exact embeddings ρ Σ,Λ : PM(S, Σ) → PM(S, Λ).

When
T is a k-rational base point, we get a fibre functor PM ls (S) → PM(k).  ) . Suppose that f : (T, Λ) → (S, Σ) and g : (V, Θ) → (T, Λ) are given, and assume the maps surject on base points. Then (f • g) * is the extension of So we obtain a natural isomorphism (f • g) * ∼ = g * • f * . These natural transformations can be seen to have the necessary compatibilities (see appendix A) to define a pseudofunctor. The last statement follows from the isomorphism The category of premotivic sheaves of F -modules is given by It follows from the discussion in appendix A that PM(S; F ) is abelian, and the natural maps PM(S, Σ; F ) → PM(S; F ) are exact.
Note that pre h i S (X, Y ) ∈ PM(S, Σ) maps to the same symbol pre h i S (X, Y ) under refinement. We denote the common value in the colimit by pre h i S (X, Y ) as well. Observe that R B provides a faithful exact embedding of PM(S, Σ) into the category Sh(S) of sheaves of F -vector spaces on S. This is compatible with refinement. So it passes to the limit. Therefore in more concrete terms, we can identify PM(S) (up to equivalence) with the directed union of subcategories Note that this lies in the subcategory Constr(S) of constructible sheaves. To be clear we always mean constructibility with respect to a stratification for the Zariski topology.
As a corollary to lemma 3.4.3.
There are a number of useful variations of this construction. Let ∆ c (S, Σ) ⊂ ∆(S, Σ) denote the full subgraph consisting of tuples (X → S, Y, i, w) where X is projective. Then set PM c (S, Σ) be the category of comodules over End ∨ (H Σ | ∆c(S,Σ) ). The category of compact motives This can be regarded as an abelian subcategory of PM(S).

Motivic Sheaves
There is one desirable property which is not evident in PM(S) which is that motivic sheaves ought to be able to be patched on a Zariski open cover. We refine PM(S) to a new category M(S) which satisfies this. We start with an abstract description of the category which characterizes it. Then we give a more concrete construction which will be useful later on. We can repackage PM(S) in the language of fibred categories [Gi, Vi]. Given a functor π : F → C, the fibre π −1 (A) over A ∈ Ob(S) is the category with objects π −1 A and morphism π −1 id A . Fibres need not behave as expected, for example fibres over isomorphic objects need not be equivalent, unless further conditions are imposed. An arrow φ ∈ M orF is cartesian if for any commutative diagram consisting of solid arrows the dotted arrow can be filled in uniquely. The functor π : F → C is fibred if any arrow of C can be lifted to a cartesian arrow of F with a specified target. It is sometimes convenient to fix a collection of specific cartesian lifts. Such a collection is called a cleavage. Given a cleavage, there is a well defined way to define a pullback functor f * : These form a pseudofunctor. Conversely, any pseudofunctor determines a fibred category. Define PM to be the category whose objects are pairs (S ∈ V ar k , M ∈ PM(S)), and morphisms are pairs (f : ). This is fibred over V ar k via the natural projection π : PM → V ar k . The categories PM(S) are just the fibres π −1 (S), and the original pseudofunctor is determined by the cleavage

Given an open cover
4. There is a functor ι : PM → M of fibred categories. This means that it fits into a commutative diagram and takes cartesian arrows to cartesian arrows. The functor ι is univeral among all such functors from PM to stacks. So that any functor PM → M ′ to a fibred category satistfying conditions (2) and (3) must factor through M in an essentially unique way, or more precisely, there is an equivalence between the categories Hom f ibcat (PM, M ′ ) and Hom f ibcat (M, M ′ ).
A concrete construction of the associated stack is given in [LM,chap 3] (although stated for groupoids, it works in general). It is a two step process. First, one constructs a prestack PM + . The objects of PM + (S) are the same as for PM(S). For Hom PM + (S) (M, N ) we take the global sections of the sheaf associated to U → Hom PM(S) (M | U , N | U ). This forces condition (2) to hold. Now construct M(S) as the category whose objects consists of descent data, i.e., collections (M i ∈ PM + (U i ), g ij ) as in (3). Given two objects (M i , after passing to a common refinement we can assume that where the last map is projection. This makes into a presheaf of coalgebras. Let E ∨ (S, Σ) denote the associated sheaf on the Zariski topology S zar . It will be convenient to generalize this a bit. Let D = {D U ⊆ ∆(U, Σ)} denote a collection of subgraphs indexed by Zariski opens. This is called a compatible collection if ι : Then for a compatible collection, we define E ∨ (D) to be the sheaf associated to the presheaf U → End ∨ (H| DU ) We refer to a compatible collection D as finite if it is the smallest collection containing a finite number of given finite graphs D Ui . Let E ∨ (D)-comod denote the category of Zariski sheaves of finite dimensional F -vector spaces with a coaction by E ∨ (D). When D is finite this is isomorphic to the category of modules over the sheaf of rings Let Coh(E(D)) denote the category of coherent i.e. locally finitely presented, modules. The colimit can be realized as a subcategory of E ∨ (S, Σ)-comod. A premotivic sheaf M in PM(S, Σ) determines an object of U → M | U of E ∨ (S, Σ)-comod, which can be seen to lie in some Coh(E(D)) and therefore in Coh(E ∨ (S, Σ)). This gives a fully faithful exact functor PM(S, Σ) → Coh(E ∨ (S, Σ)).  In view of this result, we will generally denote M ′ (. . .) simply by M(. . .) from henceforth. We also define M(S, Σ) = Coh(E ∨ (S, Σ))-comod. It will be convenient to "sheafify" the discussion of §2.3. Given a functor H # : ∆(S, Σ) → F -mod 2 compatible with restriction and such that H = p 1 • H # , we can define a category M # (S, Σ) by mimicing the procedure used to define M ′ . We have refinement of corollary 2.3.2 which follows by the same argument.

Ind objects
Let Ind-A denote the category of of Ind-objects of a category A obtained by formally adjoining filtered colimits [KS]. This is abelian, when A is [loc. cit.]. This can be given a concrete description in many cases. For example, it is well known that Ind-F -mod can be identified with F -Mod. We extend this to sheaves. Recall that an object c of an additive category is compact or finitely presented if Hom(c, −) commutes with arbitrary small coproducts. For example, a finite dimensional vector space V is seen to be compact in the category of all vector spaces, because an element of Hom(V, −) is determined by its value on a finite basis. For essentially the same reasons, we have: Proof. For any collection of sheaves F , G i , we have canonical map Injectivity can be checked on stalks. Consider the projection One checks that φ ∈ Hom(F , ⊕G i ) lies in im(κ) precisely when it has finite support in the sense that there exists a finite subset J ⊂ I such that the germs p i (φ s ) = 0 for all s ∈ S and i / ∈ J. Suppose F is constructible with respect to a necessarily finite Zariski stratification Σ. Let π σ :σ → σ denote the universal cover of a stratum. Choose bases of cardinality say n σ for each H 0 (π * F ). Then φ ∈ Hom(F , ⊕G i ) is determined by its image i.e. the map r is injective. In explicit terms, r(φ) is given by a collection of n σ sections of ⊕π * σ G i for each σ. The projections r(p j (φ)) are given by simply projecting these sections to G j . It should now be clear that φ has finite support.

Corollary 3.6.2. There is a fully faithful exact embedding of Ind-Constr(S) into Sh(S).
Proof. This follows from [KS,prop 6.3.4] and the exactness of filtered colimits.
Therefore we gave an exact faithful functor

Ind-M(S) → Ind-Constr(S) → Sh(S)
given by composition. This is also denoted by R B .

Direct Images (abstract construction)
In this section, we give a general construction of direct images. It is, unfortunately, not terribly useful in this form. Later on, we will give more explicit constructions in certain cases which will yield more information.
Set  is representable by an object rf * N . The map N → rf * N extends to a functor which is necessarily the right adjoint, cf. [N, p223]. Moreover, this is automatically triangulated by [Li,prop 3.3.8].

Cellular decompositions
The concrete realization of direct images will be based on the existence "cellular" decompositions. The use of such decompositions plays a key role in Nori's work and also [A1]. This hinges on a result of Beilinson [B] that Nori calls the basic lemma. We need a slight modification of this result. Define a map f : X → S to be equidimensional if it has a equidimensional fibres.
Proposition 4.2.1. Suppose that X → S is an equidimensional morphism of relative dimension n over a smooth base of dimension at most 1. If F is a constructible sheaf on X, then there exists an open covering {S a } of S, and dense open subsets j a : U a → X a = f −1 S a such that 1. X a − U a → S a has pure relative dimension n − 1

2.
H i Sa (X a , j a ! j * a F ) = 0 unless j = n.

H i
S (X a , j a ! j * a F ) commutes with base change Proof. When dim S = 0 this was proved in [B, lemma 3.3], so we may assume that S is a curve. The argument here is essentially the same with a couple of extra steps. Choose a relative divisor D ⊂ X such that F | X−D is locally constant. Choose projective compactifactionsX ⊂ P N andS which fit into a diagram denote the inclusions. Let j s :X s − H →X s etc. denote the restrictions of these inclusions to the fibre over s ∈ S. Then arguing as on [B, p. 36], we can see that holds for a dense open set of hyperplanes H in the dual projective spaceP N . Similarly, for each s ∈ S also holds for a dense open set of hyperplanes. Ignoring the second equation for a moment, choose an H so that (6) holds. Then we claim that holds for any s ∈ S. To see this, use (6) and the proper base change theorem, SinceM | V ∩Xs [n] is perverse, and X s is affine, we can deduce from [BBD,cor. 4.1.2] that H i (X s , j s!M | V ∩Xs ) = 0 when i < n.
By [SGA4h, thm 1.9, p 240], there is an open dense S 0 ⊂ S such that the base change isomorphism holds for for all s ∈ S 0 . Letting U 0 = f −1 (S 0 ) ∩ V gives the first step. Now choose a new hyperplane H so that (6) and (7) hold for the (finitely many) points in S − S 0 . We recycle all but one of the previous symbols j, . . ., which of course depend on the choice of H. The exception is S 1 ⊂ S which now plays the role of S 0 , i.e. it denotes the maximal open for which (9) for holds for s ∈ S 1 . We can see by an argument identical to the proof of (8) that holds for s ∈ S − S 0 . When combined with (8), we see that (9) holds at these points. So {S 0 , S 1 } is our desired cover.
We define ∆ eq (S, Σ) ⊂ ∆(S, Σ) by requiring X → S and Y → S to be equidimensional. The categories PM eq (S, Σ), M eq (S, Σ) and M eq (S) are defined by the same procedure as before by restricting H to ∆ eq . The first (respectively last two) can be viewed as subcategories of PM(S) (respectively M(S)).
Lemma 4.2.2. Let X → S be an affine equidimensional morphism to a variety of dimension at most 1. Suppose that F is a constructible sheaf on X and that is a chain of closed sets each of pure relative dimension i. There exists a filtration X 0 ⊂ X 1 ⊂ . . . X n = X such that 1. X i → S is equidimensional of pure relative dimension i.

X
where j a : X a − X a−1 → X a is the inclusion. Furthermore, these sheaves commute with base change.
Proof. This follows from the previous proposition and induction on dim X.
Definition 4.2.3. Suppose that we are given a morphismX → X of S-schemes and a filtration by closed sets We refer to (X → X,X • ) as a quasi-filtration on X, and the whole thing as a quasi-filtered X-variety.
These objects form a category QV ar S , where the morphisms φ : We say that φ covers f . A filtered variety is the special case whereX → X is the identity. Let F V ar S be the full subcategory of filtered varieties. Call a filtration satisfying the conditions of the lemma 4.2.2 (3) cellular with respect to F .
The following is a relative version of Jouanolou's trick.
such thatf is affine, and π is a bundle of affine spaces (which implies that Rπ * π * F = F for any constructible sheaf F ). Moreover,f is equidimensional if f is.
Proof. When X = P N × S,X can be taken to be product of S with the Stieffel variety St N = GL(N + 1)/(GL(1) × GL(N )) as in the proof of [Jo,lemma 1.5]. For the general case, let X ⊂X ⊂ P N × S be a relative compactification. After blowing up, we can assume thatX − X is a divisor. Then the preimageX of X in St N × S will do the job.
We say that a quasi-filtration (π :X → X,X • ) is cellular with respect to a constructible sheaf F if it is a bundle of affine spaces and ifX • is cellular with respect to π * F . Note that the first assumption implies that Rπ * π * F = F . Proposition 4.2.5. Suppose that dim S ≤ 1. Every equidimensional quasiprojective morphism X → S possesses a cellular quasi-filtration with respect to any given constructible sheaf F . Every morphism of equidimensional affine schemes over S can be refined to a morphism of cellular quasi-filtered varieties. The category of cellular quasi-filtrations of a fixed variety is connected i.e. any two objects can be connected by a chain of morphisms.
Proof. The first two statements follow immediately from lemmas 4. 2.2 and 4.2.4 The last statement is again an application of lemma 4.2.2, but it takes a few steps. First we treat the special case of cellular filtrations. Given two such filtrations X • , X ′ • , lemma 4.2.2 shows that there is there is third cellular filtration X ′′ • ⊇ X • ∪ X ′ • . Next suppose that we have a cellular quasi-filtration with underlying mapX ′ → X and that this factors through a map π :X → X satisfying Rπ * π * F = F . Then again lemma 4.2.2 shows thatX → X can be completed to a cellular quasi-filtration so thatX ′ →X is a morphism in QV ar s . The general case of quasi-filtrations on arbitraryX → X andX ′ → X can be reduced to these special cases by taking the fibre product.
Given an filtration X • ⊂ X by closed sets and a sheaf F , we have a spectral sequence (10)]). When this is cellular, this reduces to an isomorphism at E 2 . Then putting this remark together with the above results yields

Direct Images (concrete construction)
In section 4.1, a right adjoint rf * to f * was constructed in DM . However, one would really like more: is an isomorphism.
So far we have mananged to check: 2. f is projective morphism and dim Q ≤ 1.
The following lemma is crucial.
is an isomorphism.
(2) The composition Then rf * is right adjoint to f * .
Proof. It is enough to check that the compositions [M, chap IV]. Since the realizations R B are embeddings, this follows from the compatibility of η, ǫ with the usual adjunctions on the categories of sheaves.
We break the proof of the theorem into cases. We start with the case where f is a closed immersion. By 2.2.9, the map ∆(S, Σ) → M(Q) given by induces  In particular, f * is right adjoint to f * .
Proposition 4.3.5. With the previous assumptions, there is a canonical isomorphism where H j stands for the jth cohomology sheaf.
Proof. When Q is a point and Y = ∅, this was originally proved in [A1, thm 3.1]. The general case can be proved by the same method. However, a slightly cleaner alternative is to deduce it from [CM]. Since the model case (Q = pt, Y = ∅) is spelled out in detail in [C], we will be content to give the broad outline. Since both sides of the purported isomorphism are stable under base change to T , we can assume that T = S. Let L = Rf * Rg * j XY ! F . We consider two filtrations on L. The first is defined by truncations P The second F is the filtration on L associated to T • , holds. Then the cellularity of T • implies that the assumptions of [CM,prop 5.6.1] are satisfied. Therefore we have a natural filtered quasi-isomorphism where Dec(F ) is the shifted filtration associated to F . This has the property that E 1 (Dec(F )) = E 2 (F ) (c.f. [De1]). Thus it follows that there is an isomorphism of the spectral sequences associated to P and Dec(F ). The spectral sequence for P is Leray with a shift in indices, and in particular E 1 (P ) = H j Q (S, H i S (X, Y )). On the other hand, we can identify using the fact, that by assumption, H • Q (X T• , . . .) commutes with base change. Hence E 1 (Dec(F )) = E 2 (F ) = H j (K i ).
A pair of stratifications Σ of S and Λ of Q will be called f -admissible if each σ ∈ Σ is a fibre bundle over some λ ∈ Λ. Given a stratification Σ ′ of S, there exists an admissible pair (Σ, Λ) for which Σ refines Σ ′ . Any pair of stratifications can always be refined so that admissibility holds. Choose admissible stratifications Σ and Λ. Let ). Let C(S, Σ) be the so called comma category whose objects are triples where K • ∈ ObC [0,∞) (M(Q, Λ)) and M ∈ ObM(S, Σ). Morphisms are pairs K • 1 → K • 2 , M 1 → M 2 satisfying obvious compatibilities. Let C iso (S, Σ) be the full subcategory consisting of triples for which φ is an isomorphism. We can identify F 2 -mod with F -mod × F -mod. There is a faithful exact functor U 2 : C(S, Σ) → (F -mod 2 given by (K • Given a quasi-filtration (T → S, T • ) and a stratification Σ, choose base points s for (S, Σ) and t • ∈ T • . We define a functor H # : ∆(S, Σ) op → C(S, Σ) iso as follows. On objects where the differentials of the complex are connecting maps and φ is given by proposition 4.3.5.
This carries an exact faithful embedding into F 2 -mod. The categories PM # (f, T → S, T • , Σ) fibred over Q-schemes via the projection to S. We can form the associated stack M # (f, T → ST • , Σ) for the Zariski topology. This can be constructed explicitly by following the procedure outlined at the end of § 3.5. In order to simplify notation, we usually just write this as M # (T • ), when the rest of the data is understood. We let h T• (X, Y )(w) denote the object of this category associated to (X, Y, i, w). We have a functor M # (T • ) → C(S, Σ) iso , and a functor p : M # (T • ) → M(S, Σ) given as a composition of this with the functor C(S, Σ) iso → M(S, Σ) given by projection onto the second factor. From lemma 3.5.7, we obtain Fix an admisible pair Σ, Λ. By lemma 4.2.2, we obtain a cellular quasifiltration (T → S, T • ). We have a functor PM # (T • ) → C iso (S) which extends to M # (T • ). We can compose this with the projections to get functors p :

Letq denote the composition
Lemma 4.3.8. There is a functor rf * fitting into the commutative diagram This functor is independent of the choice of quasi-filtration.
where Γ is the inclusion of the graph of g. Then the adjunction map f * Rf * Rg * → Rg * is the composition of the base change map and the adjunction Rp * π * → Rp * RΓ * Γ * π * = Rg * A similar statement holds for pairs (g : X → Q, Y ).
Corollary 4.3.11. If the base change map is an isomorphism, f * Rf * Rg * → Rg * can be identified with the map Rp * π * → Rg * induced by Γ.
Lemma 4.3.12. There is a natural isomorphism R B rf * Rf * RB. Proof.
Proof. Let Σ and Λ be an f -admissible pair of stratifications. Choose a quasifiltration T • → S. Define a functor by sending (X → S, Y, i, w) to the sum of the three vertices of the diagram Note that the composition of the maps is precisely the adjunction ǫ by the above lemmas. One builds a category PM ♠ (S) = End ∨ (H ♠ )-comod. From the universal property, we have a functor h ♠ which assigns to (X → S, Y, i, w) the diagram of motives In particular, we obtain a projection PM ♠ (S) → PM(S) As above one can argue that PM ♠ (S)/ ker(p) ∼ PM(S). We can see that h ♠ factors through ker p. Thus we obtain a functor ǫ : PM(S) → M orPM(S) → M orM(S) compatible with adjunction.
Lemma 4.3.14. There is a morphism η : 1 → rf * f * compatible with the adjunction η : 1 → Rf * f * Proof. Let Σ and Λ and T • → S be as above argument. Define by sending (X → Q, Y, i, w) to the sum of vertices of the diagram This yields the η once, we observe that PM ♣ (S)/ ker(p) ∼ PM(S) where p is the natural projection.

Extension by zero
Let S ∈ V ar k and let j : S →S be a partial compactification with boundary ∂S =S − S. Suppose that Σ is a stratification ofS such that S is union of strata.  It is compatible with the usual j ! : Sh(S) → Sh(S). Suppose that π :S →S is a proper map, which is an isomorphism over S. Then we have commutative diagram This is an equivalence by the previous theorem and the relation π * j * =j * . Then we can make choices so that π * j ! =j ! alternatively we can view this as holding up to equivalence.
compatible with the base change isomorphism on realizations.
Proof. Choose a relative compactification , provided that we can show that it is well defined. First observe that we have ) as required. To see that it is independent of the choice, first observe that given a second compactification S →S, we can find a third compactification S which dominates bothS andS. For example, we can take S equal the closure of diagonal inS × QS . Thus we can assume that S =S, and so we can assume that we have a commutative diagram which induces identity on realization. So it must be an isomorphism in M(Q).
Observe thatf is necessarily equidimensional if f is. So this case can be handled by theorem 4.4.3.

Tensor products
We have a product structure on ∆(S, Σ) (and ∆ eq (S, Σ)) given by which makes it into a monoid in the category of graphs with unit (id S , ∅, 0, 0). Unfortunately, this does not immediately lead to a product on M(S, Σ). The problem has to do with the Künneth formula. To remedy this, we define a full subgraph ∆ cell (S, Σ) ⊂ ∆ eq (S, Σ) The objects of ∆ cell consist of quadruples (X → S, Y, i, w) such that X → S is affine and such that H j S (X, Y ) = 0 unless j = i. Thanks to Künneth's formula, we have a commutative diagram With this, PM cell (S) = End ∨ (H| ∆ cell (S,{S}) )-comod becomes a tensor category. This can be completed to a stack M cell (S, Σ) for the Zariski topology using the procedure of § 3.5. The tensor product extends to this. To summarize The key point is Proof. Set ∆ = ∆ eq (S, Σ), PM = PM eq (S, Σ) and PM cell = PMcell(S, Σ). Restriction gives a functor PM cell → PM which is necessarily exact and faithful. It suffices to show that this is an equivalence. We show that it is essentially surjective and full, and for this it suffices to have a right inverse up to natural equivalence. Let∆ denote the graph whose objects are sextuples (X → S, Y, i, w;X → X,X • ), such that the first four items constitutes an object of ∆, and the last two give a quasi-filtration of X in QV ar S which is cellular relative to j XY ! F . Any object of ∆ clearly lifts to∆. In particular, note that a lift of (P 1 S , {0}, 2, w) ∈ ∆ is given by There are three kinds of morphisms iñ ∆. For type I, we have a morphism of ∆ together with a morphism of QV ar S covering it. A morphism of type II is given as A morphism of type III is given as Projection gives a surjective morphism π :∆ → ∆. Proposition 4.2.5 implies that the fibres of π are connected. Thus corollary 2.2.4 implies that PM is isomorphic to the category of End ∨ (H • π) comodules.
Given (X → X, Y, i; π :X → X,X • ) in∆, by lemma 4.2.6 the complex is a complex in PM cell whose ith cohomology maps to H i S (X, Y ) (via the analogously defined R B ). Thus we get a functor PM → PM cell which is easily seen to be a right inverse to inclusion up to equivalence.
Corollary 4.5.3. When dim S ≤ 1, there is a Künneth decomposition for motives associated to objects in ∆ eq (S, {S}): Proof. This follows from the definition of the product if the objects lie in ∆ eq . In general, choose cellular quasi-filtrations (T → X, T • ) and (T ′ → X ′ , T ′ • ) as above. Their product T × S T ′ → X × S X ′ with the product filtration as the cohomology of the complex (12) for the product. The above special case of Künneth case allows one to decompose this as a tensor product of the complexes corresponding to each factor.
For objects in ∆ eq (S, {S}), we get exterior products and cup products by composing this with the restriction to the diagonal. Corollary 2.4.1 shows that these products are compatible with the standard tensor products on the categories of classical andétale local systems.

Motivic Local Systems
We assume for simplicity of exposition that dim S ≤ 1, although this should not be necessary.

Local Systems
Call an object of Γ(S) tame if it is of the form (X − D → S, E, i, w) such thatX → S is smooth and projective, and D + E is a divisor such that any intersection of components is smooth over S (we will refer to this condition as having relative normal crossings). Such a pair (X − D → S, E) is a fibre bundle, so its cohomology commutes with base change. Let ∆ tame (S) ⊂ Γ(S) be the full subgraph of tame objects. It is easy to see that ∆ tame (S) ⊂ ∆ eq (S, {S}).
Definition 5.1.1. The category of tame premotivic local systems PM tls (S; F ) = End ∨ (H| ∆tame(S) )-comod. The category of tame premotivic local systems M tls (S; F ) is the associated stack for the Zariski topology.
We note the following properties which are either immediate consequences of what has been said or easily checked. We outline the construction of Gysin maps, which will be needed later. Given a smooth subschemeȲ → S ofX transverse to D + E with relative dimension m. Set c = n − m. Then the Gysin homomorphism on cohomology can be defined simply by dualizing the restriction under Poincaré duality. However, this description is not very convenient. A better alternative is to define this via a deformation to the normal bundle as in [BFM]. LetX be the blow up ofX × A 1 alongȲ × {0}. LetỸ be the strict transform ofȲ × A 1 . Let D,Ẽ be the preimages of D, E inX. The fibre of the natural map π :X → A 1 over t = 0 is X. While the fibre π −1 0 is the union of the projectivized normal bundle p : P(N ⊕ OȲ ) →Ȳ and the blow up B ofX alongȲ . Let ). The Gysin map can then be realized as the composition of the given maps The second description yields a motivic Gysin map We can define the Gysin morphism for an arbitrary map f :Ȳ →X as the composition of the Gysin morphism associated to the inclusion of graph of Γ f ⊂Ȳ ×X followed by a Künneth projection. When, S is smooth let V M HS(S ι,an ) denote the category of rational variations of mixed Hodge structures on S ι,an , which are admissible in the sense of Steenbrink and Zucker [SZ] and Kashiwara [K]. In a nutshell, an object of this category consists of a filtered local system (V, W ) together with a compatible bifiltered vector bundle with connection (V ∼ = V ⊗O S , W, F, ∇) subject to the appropriate axioms (Griffith's transversality...). For the precise conditions, see [PS, sect. 14.4.1] or the above references. Given (X =X − D → S, E, i, 0) ∈ ∆ tame , we can construct an admissible variation as follows: This is given in [SZ], when E = ∅. The general case is easily reduced to this via the resolution where E = ∪E i is the decomposition into irreducible components. We can extend this to arbitrary objects (X =X − D → S, E, i, w) ∈ ∆ tame by tensoring the above variation with Q(w). This construction is easily checked to yield a functor ∆ tame (S) op → V M HS(S). Thus we get

Duality
The goal of this section is to prove: Theorem 5.2.1. M tls (S) is a neutral Tannakian category over F We know that this category is a tensor category over F with a tensor preserving fibre functor. What remains to be proven is that every object has a dual. By proposition 2.4.3 it is enough to construct duals for objects of the graph ∆ tame (S). We will show that where n is the relative dimension ofX → S. As first step, we note the following form of Poincaré duality.
Lemma 5.2.2. There is a pairing which is perfect in the sense that it induces an isomorphism of local systems

Proof. This follows from Verdier duality [I]
The next task is to realize the above pairing geometrically by a morphism of M tls . When D = E = ∅, we can take the cup product pairing which is induced by the diagonal embedding into the product. In general, we need to blow up the product to get a well defined diagonal. Set Y =X ×X, D 1 = D ×X, D 2 =X × D, E 1 = E ×X and E 2 =X × E. LetỸ be obtained by blowing up Y along D 1 ∩ D 2 and then along the intersection of the strict transforms of E 1 and E 2 . Let G be the exceptional divisor ofỸ → Y . Denote the strict transforms of D i , E j byD i ,Ẽ j . The diagonal embeddingX → Y extends to an embedding of d :X →Ỹ (it is not necessary to blow up X since D and E are already divisors). The image of d is disjoint fromD i ,Ẽ j and after twisting by F (n). The middle isomorphism is excision. For the last isomorphism, by projection we can reduce to the case X = P n S and then to X = (P 1 S ) n , where it follows from Künneth.
To construct δ, we dualize the above description using Gysin maps in place of pull backs: To prove (Dual), we have to establish equations (D1) and (D2). It is enough to verify these on the corresponding vector spaces H i S (X − D, E) s , H 2n−i S (X − E, D) s , and this becomes an exercise in linear algebra. If e j is a basis of the first space, and e j the dual basis of the second, then

Pure Objects
We work in M(S, Q) throughout this section. Let f : X → S be a smooth projective map of relative dimension n. Fix an embedding X ⊂ P N S . The standard generator c 1 (O(1)) ∈ H 2 (P N ) induces an isomorphism Q S (0) ∼ = h 2 S (P N S )(1). This yields a map Q S (0) → h 2 S (X)(1) by restriction. Cupping with this induces the Lefschetz operator ℓ : h i S (X) → h i+2 S (X)(1). The isomorphism follows from the usual hard Lefschetz theorem on the corresponding sheaves. Therefore we get, as usual, the Lefschetz decomposition h i . This allows us to define the Hodge involution * = * H on h * S (X) = ⊕h i S (X) by the formula in [An,. Note that the induced involution on the cohomology of a fiber H * (X s , C) coincides with the Hodge star operator with respect to the Fubini-Study metric (up to a factor and complex conjugation) [loc. cit.] Proposition 5.3.1. The algebra End(h * S (X)) is semisimple.
Proof. Set a ′ = * a t * , where a t is the transpose (c.f. [K, 1.3]). With the help of the Hodge index theorem, we see that the bilinear form trace(ab ′ ) is positive definite (compare [K, p. 381]). Then the criterion of [K, 3.13] shows that the algebra is semisimple. Proof. This is abelian and semisimple by [J2, lemma 2] and the previous proposition. It is easy to see from the constructions that it is closed under tensor product and duals.

What now?
There is still much left to be done. The current treatments of direct images and tensor products are a bit too restrictive. There is also the question of comparison with existing constructions.
André [An] has defined a semisimple Tannakian category of pure motives using his theory motivated correspondences. In [AD], the authors extended this to a relative theory. It is fairly straight forward to construct an embedding of this relative version of André's category into M pure (S). A natural question is whether this is an equivalence.
Voevodsky [VSF] has constructed a triangulated category DM gm (k) of motives over a field. Nori showed that there is a triangulated functor from DM gm (k) to the bounded derived category of his category. It remains to refine this over a base using the relative version of Voevodsky due to Cisinski-Déglise [CD].

Appendices A 2-categories
We will generally take the view that a category is the same as any other category equivalent to it. This needs some elaboration. The category Cat of all small categories is a 2-category [M]. Among other things, this means that the set of functors Hom Cat (C, D) is itself a category, where the morphisms are natural transformations. In this setting, functors are usually called 1-morphisms and natural transformations 2-morphisms. Each kind of morphism can be composed as usual. This is denoted by •. There are identities denoted by 1 X etc. There is another kind of composition for adjacent 2-morphisms, denoted here by ⋄. Given objects A, B, C, 1-morphisms F, F ′ , G, G ′ and 2-morphisms α, β as indicated below The composition β ⋄ α sits as follows It is simply given as the composition This operation is associative, and the additional identities are satisfied.
In Cat, we can either require that equations (or diagrams) hold (or commute) strictly, i.e. on the nose, or only up to a natural isomorphism. The latter is frequently the more usual occurence. Recall, for example, that categories C and D are equivalent (respectively isomorphic) if there are functors F : C → D and G : D → C such that F • G ∼ = 1 D and G • F ∼ = 1 G (respectively F • G = 1 D and G • F = 1 G ). Given a category C, a pseudofunctor F : C → Cat is an assigment of objects to objects, and morphisms to 1-morphisms, together with natural isomorphisms, ǫ c : F (1 c ) ∼ = 1 F (c) and η f,g : F (f ) • F (g) ∼ = F (f • g). These are required to satisfy certain commutivities that ensure that any two isomorphisms built from η, ǫ coincide. It suffices to check this for n ≤ 3. Contravariant pseudofunctors are simply pseudofunctors on C op .
There are number of related notions of colimit (= direct limit) of categories. We single out the notion that is most useful for us and refer to it as a 2-colimit, although "pseudo-colimit" or something like that may conform better to current usage. To simplify matters, we discuss 2-colimits in the filtered setting where things are easier (cf. [SGA4,exp VI §6]); this reference also gives a more general construction. A category D is filtered if for any two objects d 1 , d 2 , there exists an object d 3 and morphisms d 1 → d 3 , d 2 → d 3 , and for any two parallel morphisms f, g : d 1 → d 2 , there exists a morphism h : d 2 → d 3 such that hf = hg. For example, a partially ordered set is filtered precisely when it is directed. Given a pseudofunctor F : D → Cat with D filtered, the 2-colimit L = 2-lim − →d F (d) is the category whose set of objects is the disjoint union The set of morphisms from A ∈ ObF (d 1 ) to B ∈ ObF (d 2 ) is given by the filtered colimit lim − → f :d1→d3,g:d2→d3 We can see that there is a family of 1-morphisms {F (d) → L} such that for f ∈ Hom(d, d ′ ) commutes up to natural isomorphism. Moreover L would be universal in the sense that for any category L ′ with a family {F (d) → L ′ } as above, there is a unique 1-morphism L → L ′ such that the appropriate diagrams nonstrictly commute. We really want to consider 2-lim − →d F (d) only up to equivalence of categories. In practice, there may be other representations of the colimit which are more natural than the original construction.
Lemma A.0.4. Suppose that C i ⊂ C is a directed family of subcategories of a given category. Then 2-lim − → C i is equivalent to the directed union C i which the category having ObC i and M orC i as its set of objects and morphisms.
Example A.0.5. Suppose E is a coalgebra over a field F . We can express it as directed union, and therefore a 2-colimit, of finite dimensional coalgebras E = lim − → E i .
From the earlier descripition, it is not difficult to deduce the following: Proposition A.0.6. Let F be a pseudofunctor from a filtered category D to the 2-category of abelian categories. Suppose that that F In a similar vein: Proposition A.0.7. Let F be a pseudofunctor from a filtered category D to the 2-category of triangulated categories (with t-structure) and (exact) triangulated functors. Then 2-lim − →d F (d) carries the structure of a triangulated category (with t-structure) so that the functors F (d ′ ) → 2-lim − →d F (d) are triangulated (and exact).

B Comparison theorem
Let X be a C-variety. Define the site X cl with objects given by local homeomorphisms U → X an and coverings are surjective families {U i → U }. Then there is an obvious map of sites X cl → X an , which induces an equivalence of the categories of sheaves [SGA4,exp XI §4]. In particular, the cohomologies are the same. There is a canonical morphism of sites ǫ : X cl → X et which induces a map frométale to classical cohomology.
Sinceétale cohomology does not work properly for nontorsion coefficients, we start with finite coefficients, and then take the limit. Choose N > 0. A sheaf of Z/N Z-modules is constructible for either topology if there is a decomposition of X into Zariski locally closed sets, for which the restrictions are locally constant. The pullback ǫ * preserves constuctibility. The following comparison theorem is given in [SGA4,exp XVI,thm  Fix a prime ℓ. A constructible ℓ-adic sheaf is a system . . . F n → F n−1 . . . of sheaves on X et , such that each F n is a constructible Z/ℓ n Z-module and the maps induce isomorphisms F n ⊗ Z/ℓ n−1 Z ∼ = F n−1 . Standard sheaf theoretic operations can essentially be defined componentwise, and they work as expected [SGA5,SGA4h]. Given a constructible sheaf F = {F n } on X et , define on X cl . Then with this notation, the above theorem extends to the case where F is an ℓ-adic sheaf (⊗Q ℓ ) [BBD, §6.1].
C Classical t-structure for mixed Hodge modules Saito [S1,S2] has introduced a category of mixed Hodge modules 1 M HM (S). When S is nonsingular, an object M of this category consists of a filtered per-