Derived categories in algebra and topology

by J.P. May

Abstract

An analogy between the derived category of modules over a 
commutative ring and the stable homotopy category of spectra is elaborated 
to a much closer analogy between the derived category of E infinity modules 
over an E infinity algebra and the derived category of E infinity module spectra 
over an E infinity ring spectrum. In both the algebraic and topological 
contexts, these new derived categories allow one to study ``modules up to 
homotopy'' over ``commutative algebras up to homotopy'' in much the same way 
that one studies ordinary modules in classical homological algebra. There are 
many applications in algebraic topology, algebraic K-theory, and algebraic 
geometry. This expository note explains the ideas and gives a brief summary 
of the relevant definitions in both contexts. 

This paper will appear in the proceedings of the Eleventh International 
Conference on Topology, Trieste, 1993.