Let P be an abelian p-group, E a cyclic p′-group acting freely on P and k an algebraically closed field of characteristic p > 0. In this work, we prove that every self-equivalence of the stable module category of k[P oE] comes from a self-equivalence of the derived category of k[P o E]. Work of Puig and Rickard allows us to deduce that if a block B with defect group P and inertial quotient E is...

Let A and B be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence α between A and B such that α is induced by exact functors. We present a nice correspondence between indecomposable modules over A and B. As a consequence, we have the following: (1) If A is a self-injective algebra, then so is B; (2) If A and B are finite dimensional algebras over an algeb...

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, W -spaces, orth...

We construct Quillen equivalences on the Quillen model categories of rings, modules and algebras over Z-graded chain complexes and HZ-module spectra. A Quillen equivalence of Quillen model categories is the most highly structured notion of equivalence between homotopy theories. We use these equivalences in turn to produce algebraic models for rational stable model categories.

We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.