On the Formalism of Mixed Sheaves

should hold for X smooth proper, where CH(X)Q is the Chow group of X with rational coefficients, and QX denotes the constant object in D MM(X), the bounded derived category of MM(X). In this paper we give a formalism of mixed sheaves which might be useful for the construction of mixed motivic sheaves. Let k be a field embeddable in C, and V(k) the category of separated algebraic varieties over k. We choose and fix an embedding k → C. For X ∈ V(k), let X(C) denote the set of closed points of XC (:= X ⊗k C) with classical topology. For a field A of characteristic zero, let D c(X(C), A) denote the bounded derived categories of complexes of A-Modules on X(C) with constructible cohomologies, and Perv(X(C), A) the abelian category of A-perverse sheaves [3] on X(C) which is a full subcategory of D c(X(C), A). A theory of A-mixed sheaves on V(k) consists of A-linear abelian categories M(X) with A-linear forgetful functors For : M(X) → Perv(X(C), A) forX ∈ V(k) such that the categories M(X) are stable by the dual functor D, the external product ⊠, the pull-backs j by open embeddings j, and the (cohomological) direct images H f∗ by affine morphisms f, in a compatible way with the functor For, and M(X) contains a constant object AX [n] if X is smooth of pure dimension n. Moreover they should satisfy natural properties associated with these functors. Then we show that the bounded derived categories DM(X) are stable by the standard functors f∗, f!, f , f , ψg, φg,1,D,⊠,⊗,Hom in a compatible way with the corresponding functors on the underlying complexes of A-Modules by the forgetful functor, and they satisfy natural properties associated with these functors. Using these, most of the arguments in [28] [30] [31] are valid in this setting. Exception comes mainly from the Hodge theoretic description of the Picard variety of a smooth proper variety over C, which becomes quite nontrivial in other situations (e.g., for a variety over a number field). As a corollary, the notion of geometric origin can be defined as in [3]; the full subcategory M(X) of M(X) is obtained by iterating the cohomological standard functors as above to the constant object Q on Spec k and also by taking