Finite Projectivity and Contravariant Finiteness
K. Igusa, S. O. Smalo, G. Todorov · Proceedings of the American Mathematical Society · 1990
Let $\Lambda$ be an artin algebra, let $\bmod \Lambda$ be the category of finitely generated $\Lambda - {\text {modules}}$, and let $p{d_1} \bmod \Lambda$ be the subcategory of $\bmod \Lambda$ consisting of the modules of projective dimension less than or equal to one. In this paper it is proved that if the projective dimension of the injective envelope of $\Lambda$ as a $\Lambda - {\text {module}}$ is less than or equal to one, then $p{d_1}\bmod \Lambda$ is contravariantly finite in $\bmod \Lambda$. It also contains an example of finitistic projective dimension one where $p{d_1}\bmod \Lambda$ is not contravariantly finite in $\bmod \Lambda$.