This is problem 3.1.25 (page 97) in Cohen-Macaulay Rings by Bruns and Herzog. The direction I am interested in is the following. Let $R$ be a Gorenstein local ring and $M$ a finite $R$-module. If the injective dimension of $M$ is finite, then prove that the projective dimension of $M$ is finite. I am interested […]

Here is question 18.8 of Matsumura’s Commutative Ring Theory. It asks whether the rings $k[[t^3,t^4,t^5]]$, $k[[t^4,t^5,t^6]]$ are Gorenstein. I got that 1) is not Gorenstein, but 2) is Gorenstein (by computing the socle). Just wanted to check if I am correct. I don’t need the answer necessarily, a yes or a no will suffice. Thanks.

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