Let $(R,m,k)$ be a local Noetherian ring, $M$ a finitely generated $R$-module. How can I prove that $M$ has finite injective dimension if and only if it has a $\omega_R$-resolution? ($\omega_R$ is the canonical module, and an $\omega_R$-resolution is a resolution looks like this $0\rightarrow \omega_R^{r_p}\rightarrow\cdots\rightarrow\omega_R^{r_0}\rightarrow M\rightarrow0$.)

(This is exercise 3.3.28(b) of Bruns-Herzog; part (a) asks to prove that a Maximal Cohen-Macaulay module with finite injective dimension is a direct sum of copies of $\omega_R$. I don't know if we need to use this result, and if this is the case how can we prove it? I was thinking of reducing module regular elements because in the Artinian case the result is true but I don't know how to conclude.)


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