2020/01/23 by Schäppi, Daniel
#14C15 #18E30 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2001.08567
Given a rigid tensor-triangulated category and a vector space valued homological functor for which the Künneth 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 ℓ-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 ℓ-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.