Relative homological algebra and Waldhausen $K$-theory

Abstract

We study the question of the existence of a Waldhausen category on any (relative) abelian category in which the contractible objects are the (relatively) projective objects. The associated K-theory groups are "stable algebraic G-theory," which in degree zero form a certain stable representation group. We prove both some existence and nonexistence results about such Waldhausen category structures, including the fact that, while it was known that the category of R-modules admits a model category structure if R is quasi-Frobenius, that assumption is required even to get a Waldhausen category structure with cylinder functor-i.e., Waldhausen categories do not offer a more general framework than model categories for studying stable representation theory of rings. We study multiplicative structures on these Waldhausen categories, and we relate stable algebraic G-theory to algebraic K-theory and we compute stable algebraic G-theory for finite-dimensional quasi-Frobenius nilpotent extensions of finite fields. Finally, we show that the connective stable G-theory spectrum of F p n [x]/x p n is a complex oriented ring spectrum, partially answering a question of J. Morava about complex orientations on algebraic K-theory spectra.