Motives, numerical equivalence, and semi-simplicity

Integral motives and special values of zeta functions

“…The hypotheses imply that the Künneth projectors are algebraic for = p (see 4.7), and when we use them to modify the commutativity constraints (see 4.6), the resulting category M(k; Q; S) is Tannakian [26]. In any Tannakian category over a field, there is a good notion of the rank of an object X (defined to be the trace of the identity map on X; see [47, I, 5.1.4]), and for any fibre functor ω with values in the vector spaces over a field and any morphism α : X → Y in the category, dim(ω(X)) = rank(X), rank(ω(α)) = rank X − rank Ker(α).…”

“…For definiteness, take M(k; Q) to be the category of isomotives based on the smooth projective varieties over k whose Künneth projectors are algebraic -the correspondences are the numerical equivalence classes of algebraic classes. This category is semisimple [26] and Tannakian [8], and when we assume that numerical equivalence coincides with homological equivalence, it admits canonical l-adic fibre functors for each l. 1 The category M(k; Z) that we construct is noetherian, abelian, and, at least when k is finite and the Tate conjecture holds, Tannakian.…”

On a multiplicative version of Bloch’s conjecture

“…Thus, using Jannsen's semisimplicity result [15], the motives (X, π i ) and (X, Π n,0 ) are contained in a full semisimple abelian subcategory M 0 of the category M hom of motives with respect to homological equivalence (for M 0 ⊂ M hom one can take the subcategory generated by varieties that are known to satisfy the Lefschetz standard conjecture). Hence, we get a decomposition…”

The motivic Thom isomorphism

“…Grothendieck's original category of 'pure' motives, constructed from smooth projective varieties, is (in some generality [28]) semisimple, but categories of motives built from more general (non-closed) varieties admit nontrivial extensions. The (derived) category DM T Q (k) of mixed Tate motives can be defined as the smallest tensor triangulated subcategory of DM Q (k) containing the Tate objects.…”