Finite projectivity and contravariant finiteness

Igusa, K.; Smalø, S. O.; Todorov, G.

doi:10.1090/S0002-9939-1990-1027094-0

Finite projectivity and contravariant finiteness
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by K. Igusa, S. O. Smalø and G. Todorov PDF

Proc. Amer. Math. Soc. 109 (1990), 937-941 Request permission

Abstract:

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 $\bmod \Lambda$.

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