Canonical Modules

Question

Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2-yz}$? I am not necessarily looking for an explicit description of the canonical module, but I would appreciate any and all suggestions for describing its structure.

Answer 1

As Graham points out, this ring is Gorenstein so the canonical module is isomorphic to the ring itself. For most hypersurfaces, this is all you can say. However, I think in this case one can say slightly more.

The ring is also toric $k[x,y,z]/(x^2 - yz) = k[ab, a^2, b^2]$. Thus we have a canonical way to identify the canonical module/divisor.

Recall that $K_X = -\sum \text{[torus invariant prime divisors]}$. For this ring, we are looking for height 1 primes that are toric (monomial). There are two $(ab, a^2) = (x,y)$ and $(ab, b^2) = (x,z)$. The negative sum of the corresponding divisors just corresponds to the intersection of the two ideals. In this case, we get $(ab, a^2) \cap (ab, b^2) = (ab) = (x)$. So the ideal $(x)$ is the canonical module.

Determinantal rings also have explicit canonical choices for canonical modules if I recall correctly. Graded rings have graded canonical modules which have some canonical choice of degree.