Graded-Tannakian categories of motives

Abstract

Given a rigid tensor-triangulated category and a vector space valued homological functor for which the K\"{u}nneth isomorphism holds, we construct a universal graded-Tannakian category through which the given homological functor factors. We use this to (unconditionally) construct graded-Tannakian categories of pure motives associated to a fixed Weil cohomology theory, with a fiber functor realizing the given cohomology theory. For $\ell$-adic cohomology and a ground field which is algebraic over a finite field, this category is Tannakian. In this case, we obtain in particular motivic Galois groups which act naturally on $\ell$-adic cohomology without assuming any of the standard conjectures. We show that these graded-Tannakian categories are equivalent to Grothendieck's category of pure motives if the standard conjecture D holds.