Derived categories and motives

Abstract

Ab stract. 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