  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.
