And K - Theory

In this paper we show how to write any variety over a field of characteristic zero as the difference of two pure motives, thus answering a question asked by Serre (in arbitrary characteristic, [Se] p.341). The Grothendieck theory of (pure effective Chow) motives starts with the category V of smooth projective varieties over a field k. A motive is a pair (X, p), where X is in V and p is a projector in the ring of algebraic correspondences from X to itself (algebraic cycles on X ×X modulo linear equivalence). One can add two motives Let K 0 (M) be the abelian group associated to this monoid. Assuming the field k has characteristic zero, for any variety X over k (i.e. any reduced scheme of finite type over k) we define a class [X] in K 0 (M), which is characterized by the following two properties: if X is connected and lies in V, [X] is the class of the motive (X, ∆ X), where ∆ X is the diagonal in X × X; if Y is a closed subset in X, with its reduced scheme structure, the following identity holds in K 0 (M): (0.1) [X] = [Y ] + [X − Y ]. This result (Theorem 4) is derived from a stronger one. Namely, to any variety X we associate a cochain complex W (X) in the additive category of complexes of motives, which is well-defined up to canonical homotopy equivalence. Our construction of W (X) and the proof of its properties (Theorem 2) use higher algebraic K-theory and, more specifically, the Gersten complexes of schemes. These complexes are made out of the K-theory of all residue fields of a given scheme, and among their homology groups are precisely the Chow groups of algebraic cycles modulo linear equivalence (see Section 1.1). To prove Theorem 2, we first extend Manin's identity principle [M1], by showing that a complex C. of varieties is con-tractible as a complex of motives if and only if a certain family of Gersten complexes associated to C. are acyclic (Theorem 1). We then use a variant of the theory of cohomological descent which applies to Gersten complexes (and to K ′-theory) of simplicial schemes, and was developped by the first author in [G2] (Proposition 1). Here the proper surjective maps used in the conventional theory of cohomological descent ([SD], [D]) are replaced by envelopes …