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Let $\mathcal{E}$ be a weakly idempotent complete exact category with enoughinjective 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 takingdirect sums and summands. In this paper, $\mathcal{E}$ is equipped with thestructure of a prefibration category with cofibrant replacements. As acorollary, we show, using the results of Demonet and Liu in \cite{DL}, that thecategory of finite presentation modules on the costable category$\overline{\mathcal{M}}$ is a localization of $\mathcal{E}$. We also deducethat $\mathcal{E} \to \mathrm{mod}\overline{\mathcal{M}}$ admits a calculus offractions up to homotopy. These two corollaries are analogues for exactcategories 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 ofprefibration category to the structure of a model category (see the article ofPalu in \cite{Palu} for the case of triangulated categories). This last resultapplies in particular when $\mathcal{E}$ is any of the Hom-finite Frobeniuscategories appearing in relation to cluster algebras.

Model categories structures from rigid objects in exact categories