Relative homological algebra and Waldhausen $K$-theory

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 $\mathbb{F}_{p^n} [x] / x^{p^n}$ is a complex-orientable ring spectrum, partially answering a question of J. Morava about complex orientations on algebraic $K$-theory spectra.

Keywords

Waldhausen category, algebraic $K$-theory, quasi-Frobenius ring, relative homological algebra, complex oriented ring spectrum

2010 Mathematics Subject Classification

13C60, 18G25, 19D55

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Received 4 November 2015

Received revised 29 June 2017

Published 3 January 2018