ALGEBRAS AND MODULES IN MONOIDAL MODEL CATEGORIES

Abstract

In recent years the theory of structured ring spectra (formerly known as A$_{\infty}$- and E$_{\infty}$-ring spectra) has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra. In this paper we provide a general method for constructing model category structures for categories of ring, algebra, and module spectra. This provides the necessary input for obtaining model categories of symmetric ring spectra, functors with smash product, $\Gamma$-rings, and diagram ring spectra. Algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra. 1991 Mathematics Subject Classification: primary 55U35; secondary 18D10.