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Let $(R,\frak{m})$ be a hypersurface (i.e., $R=Q/(f)$, where $Q$ is a regular local ring, $0\not=f\in Q$). If $M$ is a MCM $R$-module, is it possible for the injective dimension of $M$ over $R$ to be $\infty$? What about the Gorenstein injective dimension in this case?

Since a noetherian local ring is regular iff it has finite global dimension = max projective dimension of all finite modules = max injective dimension of all modules (Matsumura, p.155), over a regular local ring we have that of course $M$ will have finite injective (and hence Gorenstein injective dimension). But what about over $R$?

Thank you!

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If $M$ is not free it will have infinite injective dimension. This is because $R$ is Gorenstein (since a hypersurface), so the projective dimension of a finitely generated module is finite iff its injective dimension is. By Auslander-Buchsbaum, finite projective dimension for a MCM module is equivalent to freeness.

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  • $\begingroup$ Of course, thanks. For completeness, I'll also add that in a Gorenstein ring $R$, the Gorenstein injective dimension of any $R$-module is always finite (cf. 6.2.7 of Christensen's "Gorenstein Dimensions"). $\endgroup$
    – math-grad
    Commented May 22, 2014 at 15:57

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