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Keywords:
finitistic dimension; Gorenstein projective module; endomorphism algebra
finitistic dimension; Gorenstein projective module; endomorphism algebra
Summary:
Let $A$ be a CM-finite Artin algebra with a Gorenstein-Auslander generator $E$, $M$ be a Gorenstein projective $A$-module and $B = {\rm End}_A M$. We give an upper bound for the finitistic dimension of $B$ in terms of homological data of $M$. Furthermore, if $A$ is $n$-Gorenstein for $2 \leq n < \infty $, then we show the global dimension of $B$ is less than or equal to $n$ plus the $B$-projective dimension of ${\rm Hom}_A(M, E).$ As an application, the global dimension of ${\rm End}_A E$ is less than or equal to $n$.
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