Let $R$ be a Cohen-Macaulay local ring with canonical module $\omega_R$ and $I$ an ideal of $R$ of height $g$.
Write $\omega_{R/I} = \operatorname{Ext}^g_R(R/I, \omega_R)$. It is well-known that whenever $R/I$ is Cohen-Macaulay, the module $\omega_{R/I}$ is the canonical module of $R/I$. In particular, $\omega_{R/I}$ is a faithful $R/I$-module; in other words, its annihilator is $\operatorname{ann}_R(\omega_{R/I}) = I$.
I wonder under what conditions (of course, weaker than $R/I$ Cohen-Macaulay; e.g., $I$ is unmixed) we have $\operatorname{ann}_R(\omega_{R/I}) = I$.
A possibility could be to investigate when $\omega_{R/I}$ is a semidualizing module over $R/I$ (is it?), since such modules are faithful.
Thank you in advance.
Best Answer
This happens if and only if $R/I$ is unmixed and equidimensional. In general, there is an injection $R/J(R )\to \operatorname{Hom}_R(\omega,\omega)$ where $J(R)$ is the maximum of the set $\{ I: I \text{ an ideal of } R \text{ with } \dim I<n\}$ and $J(R)=0$ if and only $R$ is unmixed and equidimensional. See Theorem 12.1.15, [1].
[1] Brodmann, M. P. (Markus P.), and R. Y. Sharp. Local Cohomology : an Algebraic Introduction with Geometric Applications. 2nd ed. Cambridge: Cambridge University Press, 2013. Print.