What is a generating set of the canonical module? Let $R = k[x_1,...,x_n]$ be a standard graded polynomial ring over a field $k$. Let $I$ be an ideal of $R$ such that $R/I$ is Cohen-Macaulay. Then the canonical module $\omega_{R/I}$ can be identified with an ideal in R, say $J$. For any such identification, $\omega_{R/I}$ is either an ideal of height 1 or equals $R/I$ (by Proposition 3.3.18, "Cohen-Macaulay rings" Bruns-Herzog). 
Now, my question is that what will be a generating set for $J$?  The proof of proposition 3.3.18 does not say anything about the generating set. Any ideas/hints are welcome.
 A: Understanding a minimal generating set of the canonical module is an arduous task; even the minimal number of generators of the canonical module, which is the type of $S/I$, contains a great deal of information. Suppose $S$ is regular and let's say either local or standard graded. One can compute a presentation matrix for the canonical module provided one knows the resolution of $S/I$ over $S$ (of course one will not know this in general).

Theorem: Let $I$ be an ideal in $S$ and let $R=S/I$.  Suppose $R$ is Cohen-Macaulay, and let
$$F_{\bullet}:0 \rightarrow S^{\beta_t^S(R)} \xrightarrow{A_t} S^{\beta_{t-1}^S(R)} \rightarrow \cdots \rightarrow  S^{\beta_2^S(R)} \xrightarrow{A_2} S^{\beta_1^S(R)} \xrightarrow{A_1} S \rightarrow 0$$ be the minimal free resolution of $R$ over $S$.  Then $w_R$ has an $R$-presentation of the form $$R^{\beta_{t-1}^S(R)} \xrightarrow{A^T_t} R^{\beta_t^S(R)} \rightarrow{} w_R \rightarrow 0.$$


Proof: By Corollary 3.3.9 in "Cohen-Macaulay Rings" by Bruns and Herzog, $\operatorname{Hom}_S(F_{\bullet},S)$ is a minimal free resolution of $\omega_R$. In particular,
$$S^{\beta_{t-1}^S(R)} \xrightarrow{A^T_t} S^{\beta_t^S(R)} \rightarrow{} w_R \rightarrow 0$$
is a minimal $S$-presentation of $\omega_R$. The result follows from applying $- \otimes_S R$ to this presentation.

There are also a number of cases where we have a deeper understanding of the canonical module.
One such case comes from the theory of linkage (also known as the theory of liaison).  Linkage can be considered in more generality, but for the case you're interested in the following will suffice.

Definition: Let $S$ be either a standard graded $k$-algebra or a local ring with residue field $k$, and further assume $S$ is regular (so in the graded case $S$ is a polynomial ring). Two ideals $I$ and $J$ of height $g$ are said to be (directly) linked if there is a regular sequence $\underline{\alpha}=\alpha_1,\dots,\alpha_g \subseteq I \cap J$ such that $I=\underline{\alpha}:J$ and $J=\underline{\alpha}:I$.


Theorem: Suppose $S$ is as in the definition above. If $I$ is unmixed and $I$ and $J$ are linked (by the regular sequence $\underline{\alpha}$), then $R/I$ is Cohen-Macaulay if and only if so is $R/J$.  Furthermore, in this case, $\omega_{R/I} \cong J/\underline{\alpha}$ and $\omega_{R/J} \cong I/\underline{\alpha}$.

This theorem appears originally in Peskine and Szpiro's Liaison des variétés algébrique, which historically was the jumping off point for linkage theory; see Proposition 1.3 and the ensuing Remarques. I think The structure of linkage gives a good overview of this theory.
Other cases where it is well understood include determinantal rings and Veronese rings.
