# On the existence of finitely generated modules with finite injective dimension

Assume $R$ is a commutative local Noetherian ring. It is known that if there is a finitely generated module with finite injective dimension then $R$ is Cohen-Macaulay. My question is:

if $R$ is CM is there a finitely generated module with finite injective dimension?

I think I read the construction of such module in Bruns-Herzog but I cannot remember it nor I can find it on the book. Of course if the ring is a homomorphic image of a Gorenstein ring the you can just take the canonical module, but in general is there such module?

## 2 Answers

Hint. Take a system of parameters in $R$, say $x_1,\dots,x_d$, and $S=R/(x_1,\dots,x_d)$. Now set $M=E_S(k)$, where $k$ is the residue field of $R$.

Maybe part of Bruns-Herzog you've read is:

Note that this contains user26857's answer, too.