On contravariant finiteness of subcategories of modules of projective dimension $\leq I$

Let $\land$ be an artin algebra. This paper presents a sufficient condition for the subcategory $\mathcal {P}^{i}( \land )$ of $\mod \land$ to be contravariantly finite in $\mod \land$, where $\mathcal {P}^{i}( \land )$ is the subcategory of $\mod \land$ consisting of $\land$--modules of projective dimension less than or equal to $i$. As an application of this condition it is shown that $\mathcal {P}^{i}( \land )$ is contravariantly finite in $\mod \land$ for each $i$ when $\land$ is stably equivalent to a hereditary algebra.