Derived Categories in Algebra and Topology

J. P. MAY I will give a philosophical overview of some joint work with Igor Kriz (in algebra), with Tony Elmendorf and Kriz (in topology), and with John Greenlees (in equivariant topology). I will begin with a description of some foundational issues before saying anything about the applications. This is not the best way to motivate people, but I must explain the issues involved in order to describe what we have done. Let me just say that the emphasis I shall give to an analogy between algebra and topology is not just an expository device. The algebraic work that I will describe both illuminates the deeper topological theory and has applications to algebraic geometry. We begin by displaying an analogy that is familiar to topologists. It is the starting point of our work. ALGEBRA TOPOLOGY a commutative ring k the sphere spectrum S differential graded k-modules spectra tensor product smash product internal hom Hom(X, Y ) function spectrum F (X, Y ) dual DX = Hom(X, k) dual DX = F (X,S) projective k-module CW spectrum finitely generated projective finite CW spectrum Hom(X, Y )⊗ E ∼= Hom(X, Y ⊗ E) F (X, Y ) ∧ E ' F (X, Y ∧E) DX ⊗ E ∼= Hom(X,E) DX ∧E ' F (X,E) (X or E fin. gen. projective) (X or E a finite CW spectrum) hyperhomology Eq(X) ≡ πq(X ∧E) hypercohomology E(X) ≡ π−qF (X,E) Note that the right column already encodes the important topological theory of Spanier-Whitehead duality: if X is a finite CW spectrum, then E∗(DX) ∼= E−∗(X).