Injective resolutions of the module of Kähler differentials

Question

Let $k$ be a field, $A=k[x_1,\dots,x_n]/I$ an affine algebra and $\Omega_{A|k}$ the $A$-module of Kähler differentials. By abstract nonsense there exists an injective resolution $\mathcal I$ of $\Omega_{A|k}$. Are there known constructions for $\mathcal I$ that one can actually work with (i.e. do calculations)? Already the case of a singular hypersurface would be interesting.

Answer 1

One approach is the Cousin complex. See Rodney Sharp, "The Cousin complex for a Module over a Commutative Noetherian Ring," Math. Z., 1969. In "Gorenstein Modules" (Math. Z., 1970), Sharp defines a Gorenstein module as one whose Cousin complex is an injective resolution, and in Theorem 3.6 of that paper he proves some reasonable equivalent conditions to being a Gorenstein module. So you can look at Sharp's conditions to try to check whether the Kahler 1-forms are Gorenstein in this sense, in the cases you care about. I imagine the answer depends on what kind of singularity your hypersurface has. (At least this is the case for coefficients in R rather than in the Kahler 1-forms: in that case, R is a Gorenstein R-module precisely when the singularity is Gorenstein.)

As I recall, when the Cousin complex is an injective resolution, it's also a MINIMAL injective resolution, which makes it fairly tractable for explicit calculation.