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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 in knowing how to do this exercise by using results proved upto this point in the book.

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Use induction on $d=\dim R$.

If $d=0$, then $\text{inj}\dim M=0$, so $M$ is injective. Take a short exact sequence $$0\longrightarrow N\longrightarrow F\longrightarrow M\longrightarrow 0$$ with $F$ free of finite rank, notice that $F$ is injective and then $\text{inj}\dim N<\infty$. Furthermore, $\text{inj}\dim N=0$ and then the short exact sequence is split. It follows that $M$ is projective.

If $d>0$, then choose $x\in m$ ($m$ is the maximal ideal of $R$) which is nonzerodivisor on $R$ and on $N$, where $N$ (in this case) is the $d$th syzygy of $M$. (One can do this, since from B&H, exercise 2.1.26, one knows that $N$ is $0$ or maximal CM, so $m$ cannot be contained in the set of zerodivisors of $N$, otherwise $\text{depth } N=0$ which implies $d=0$, contradiction.) Notice again that $\text{inj}\dim N<\infty$ and now specialize all by $x$ and apply the induction hypothesis.

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  • $\begingroup$ Can you indicate how one could do exercise 2.1.26? $\endgroup$ – messi Jun 6 '12 at 12:26
  • $\begingroup$ I just saw this comment from you, so i will post this exercise as a different problem. The above answer from you is very good, and i would like to vote it up but i dont know how to do that. $\endgroup$ – messi Jun 17 '12 at 8:58

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