Gorenstein AC-projective complexes

TL;DR

This paper constructs a new model structure on chain complexes over any ring, identifying Gorenstein AC-projective complexes as cofibrant objects, and explores their properties and precovers in various ring contexts.

Contribution

It introduces a cofibrantly generated abelian model structure on chain complexes with Gorenstein AC-projective complexes as cofibrant objects, extending Gorenstein projective theory.

Findings

Gorenstein AC-projective complexes are cofibrant in a new model structure.

Every chain complex has a Gorenstein AC-projective precover.

The model structure is finitely generated over Ding-Chen rings.

Abstract

Let $R$ be any ring with identity and Ch($R$) the category of chain complexes of (left) $R$-modules. We show that the Gorenstein AC-projective chain complexes are the cofibrant objects of an abelian model structure on Ch($R$). The model structure is cofibrantly generated and is projective in the sense that the trivially cofibrant objects are the categorically projective chain complexes. We show that when $R$ is a Ding-Chen ring, that is, a two-sided coherent ring with finite self FP-injective dimension, then the model structure is finitely generated, and so its homotopy category is compactly generated. Constructing this model structure also shows that every chain complex over any ring has a Gorenstein AC-projective precover. These are precisely Gorenstein projective (in the usual sense) precovers whenever $R$ is either a Ding-Chen ring, or, a ring for which all level (left) $R$-modules have finite projective dimension. For a general (right) coherent ring $R$, the Gorenstein AC-projective complexes coincide with the Ding projective complexes and so provide such precovers in this case.