Hi,

Despite being nothing but the dual notion of projective resolution, injective resolutions seem to be harder to grasp. For example, the general form of Poincaré-Lefschetz duality given in Iversen's *Cohomology of sheaves* (p. 298) uses an injective resolution of the coefficient ring k (which is assumed to be Noetherian) as a k-module, a notion whose projective equivalent is rather meaningless.

My question is: do we have explicit injective resolutions of some simple (but not principal) rings (as modules over themselves) like the polynomial ring $\mathbb{C}[X,Y]$ or its Laurent counterpart $\mathbb{C}[X,Y,X^{-1},Y^{-1}]$? By general dimension arguments, some short resolutions exist, but I'm unable to find them explicitly.

Thanks!