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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 1

### Models for mock homotopy categories of projectives

Pages: 247 – 263

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n1.a13

#### Author

#### Abstract

Let $R$ be a ring and $\mathrm{Ch}(R)$ the category of chain complexes of $R$-modules. We put an abelian model structure on $\mathrm{Ch}(R)$ 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.

#### Keywords

abelian model structure, chain complex of projectives, triangulated category, quasi-coherent sheaf, recollement

#### 2010 Mathematics Subject Classification

18E35, 18G25, 55U35