Model categories structures from rigid objects in exact categories

Abstract

Let $\mathcal{E}$ be a weakly idempotent complete exact category with enough injective and projective objects. Assume that $\mathcal{M} \subseteq \mathcal{E}$ is a rigid, contravariantly finite subcategory of $\mathcal{E}$ containing all the injective and projective objects, and stable under taking direct sums and summands. In this paper, $\mathcal{E}$ is equipped with the structure of a prefibration category with cofibrant replacements. As a corollary, we show, using the results of Demonet and Liu in \cite{DL}, that the category of finite presentation modules on the costable category $\overline{\mathcal{M}}$ is a localization of $\mathcal{E}$. We also deduce that $\mathcal{E} \to \mathrm{mod}\overline{\mathcal{M}}$ admits a calculus of fractions up to homotopy. These two corollaries are analogues for exact categories of results of Buan and Marsh in \cite{BM2}, \cite{BM1} (see also \cite{Be}) that hold for triangulated categories. If $\mathcal{E}$ is a Frobenius exact category, we enhance its structure of prefibration category to the structure of a model category (see the article of Palu in \cite{Palu} for the case of triangulated categories). This last result applies in particular when $\mathcal{E}$ is any of the Hom-finite Frobenius categories appearing in relation to cluster algebras.