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Gorenstein dimension is a kind of homological invariants of finitely generated modules.

Are the following results true?:

(1). Let $R$ be a Noetherian local ring and $M$ be a finitely generated $R$-module with finite projective dimension. If $Ext_{R}^{1} (M,R) =0$, then $M$ is projective.

(2). Let $R$ be a Noetherian local ring with unit and $M$ be a finitely generated $R$-module with finite Gorenstein dimension. If $Ext_{R}^{1} (M,R) =0$, then $M$ is projective.

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(1): let $R$ be a regular local ring of dimension at least $2$, and $M=k$ be the residual field (not projective). Then $M$ has finite projective dimension (because regularity by https://stacks.math.columbia.edu/tag/065U, eg Prop 10.110.1), and $\mathrm{Ext}^1_R(M,R)=0$ because $R$ has depth at least $2$ (by https://stacks.math.columbia.edu/tag/00LE, Prop 10.72.5, then https://stacks.math.columbia.edu/tag/00NN, Lemma 10.106.3).

(2): I expect the same kind of counter-example, but I can’t find a good reference for Gorenstein dimension in Stacks.

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  • $\begingroup$ The same counterexample works for (2). Any module of finite projective dimension also has finite Gorenstein dimension. In fact, the Gorenstein dimension and projective dimension are equal when they are both finite. An advantage of Gorenstein dimension is that it is finite more frequently; e.g. every module over a Gorenstein ring has finite Gorenstein dimension. $\endgroup$ Nov 25, 2021 at 22:29

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