Derived categories and motives

Abstract

We relate derived categories of modules over rational DGA's to categories of comodules over associated Hopf algebras, and we explain how this implies the equivalence of definitions of mixed Tate motives proposed by Bloch and Deligne. We also describe an approach to integral mixed Tate motives in terms of the derived categories of modules over certain E ∞ algebras. We first explain some new differential homological algebra-alias rational homotopy theory-over a field of characteristic zero and then use it to show the equivalence of two proposed definitions of mixed Tate motives [8, 9, 5] in algebraic geometry. One of these has been proven to admit Hodge andétaleandétale realizations, but is intrinsically restricted to the rational world. The other can be linked up to a speculative definition of integral or modular mixed Tate motives, and we end by explaining some new differential homological algebra over arbitrary commutative rings that is necessary to make sense of this approach to mixed Tate motives. Let A be a commutative differential graded and "Adams graded" k-algebra, abbreviated DGA, where k is a field of characteristic zero. Thus A is bigraded via k-modules A q (r), where q ∈ Z and r ≥ 0. We assume that A q (r) = 0 unless 2r ≥ q. The differential and product behave as follows with respect to the gradings: d : A q (r) → A q+1 (r) and A q (r) ⊗ A s (t) → A q+s (r + t). We assume that A has an augmentation ε : A → k. Write H q (A)(r) for the cohomology of A in bidegree (q, r). We also assume throughout that A is cohomologically connected, in the sense that H q (A)(r) = 0 if q < 0,