I'm studying local cohomology and the canonical module of a local Cohen-Macaulay ring $R$ is very important due to local duality and its consequences (as non-vanishing of $d$-th local cohomology of an $R$-module $M$ where $d=\dim(M)$).
The canonical module can be defined as a finitely generated $R$-module whose Matlis dual is isomorphic to $d$-th local cohomology of $R$ where $d=\dim(R)$. (Some books call it dualizing module.) I know that such a module is unique (up to isomorphism), maximal Cohen-Macaulay of type $1$ and of finite injective dimension. I also know that the converse holds.
On the other hand, how to proceed if $R$ does not admit a canonical module? Are many results lost on about local cohomology? Also I would like to see some example of Cohen-Macaulay ring without canonical module. Having in mind that a Cohen-Macaulay ring admits a canonical module if, and only if, its an epimorphic image of a Gorenstein ring, I tried to get one, but I couldn't.
Thanks for any advise.
Best Answer
Please see the example 6.1 of the following paper:
"A few examples of local rings I", Jun-ichi Nishimura.