Relative homological algebra and Waldhausen \(K\)-theory (Q1707280)

This paper studies the existence of Waldhausen (abelian) categories where the associated \(K\)--theory groups form stable \(G\)-theories, and provides applications. Here the word ``stable'' has its origin to stable module categories. Recall that if \(R\) is a ring, the stable category \(\underline{\mathrm{Mod}}(R)\), has the same objects as \({\mathrm{Mod}}(R)\), and that two parallel morphisms are equivalent if their difference factors through a projective module. The paper is written in the language of relative homological algebra: the author considers an abelian category \(\mathcal{C}\), together with an ``allowable class'' of short exact sequences \(E\). Then \(E\)-projectives, \(E\)-injectives and \(E\)-stable equivalences are introduced. The author provides sufficient conditions imposed on an abelian category \(\mathcal{C}\), equipped with two allowable classes \(F\subseteq E\), in order for it to be a Waldhausen category, where the cofibrations are the \(F\)-monomorphisms and the weak equivalences are the \(E\)-stable equivalences. He also provides sufficient conditions on such a category for Waldhausen's fibration theorem to hold [\textit{F. Waldhausen}, Lect. Notes Math. 1126, 318--419 (1985; Zbl 0579.18006)]. The main result is as follows: If \(\mathcal{C}\) is a quasi-Frobenius abelian category with enough projectives and functorially enough injectives, then \(\mathcal{C}\) admits the structure of a Waldhausen category where the cofibrations are the monomorphisms and the weak equivalences are the stable equivalences. This Waldhausen structure also satisfies the extension and saturation axioms and admits a cylinder functor which satisfies the cylinder axiom. Some applications include a multiplicative structure on the stable \(G\)-theory spectrum of a finite dimensional co-commutative Hopf algebra over a finite field, and a computation of the stable \(G\)-theory of certain truncated polynomial algebras, based on work of \textit{I. Madsen} [in: Current developments in mathematics, 1995. Lectures of a seminar, held in Boston, MA, USA, May 7-8, 1995. Cambridge, MA: International Press. 135--189 (191--321, preliminary version 1994) (1995; Zbl 0876.55004)].