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!



Is it possible for a MCM module over a hypersurface to have infinite injective dimension?
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.


