Models for mock homotopy categories of projectives

TL;DR

This paper constructs a model structure on chain complexes over a ring or scheme that captures the mock homotopy category of projectives, linking it to contraderived categories and generalizing via cotorsion pairs.

Contribution

It introduces a new abelian model structure on chain complexes whose homotopy category models the mock homotopy category of projectives, extending to schemes and other cotorsion pairs.

Findings

Model structure on Ch(R) with homotopy category equivalent to K(Proj)

Homotopy category coincides with contraderived category of Positselski

Reconstruction of Murfet's recollement and generalizations with cotorsion pairs

Abstract

Let $R$ be a ring and Ch($R$) the category of chain complexes of $R$-modules. We put an abelian model structure on Ch($R$) whose homotopy category is equivalent to $K(Proj)$, the homotopy category of all complexes of projectives. However, the cofibrant objects are not complexes of projectives, but rather all complexes of flat modules. The trivial objects are what Positselski calls contraacyclic complexes and so the homotopy category coincides with his contraderived category. We in fact construct this model on the category of chain complexes of quasi-coherent sheaves on any scheme $X$ admitting a flat generator. In this case the homotopy category recovers what Murfet calls the mock homotopy category of projectives. In the same way we construct a model for the (mock) projective stable derived category, and we use model category methods to recover the recollement of Murfet. Finally, we consider generalizations by replacing the flat cotorsion pair with other complete hereditary cotorsion pairs in Grothendieck categories.