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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.


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$\newcommand{\C}{\mathbb C} $I think this is OK. The first step is the inclusion of $\C[X,Y]$ into its fraction field which is $\C(X,Y)$. For each irreducible polynomial $f$ (normalised so that the top degree monomial for some ordering is $1$) we map $\C(X,Y)$ to $\C(X,Y)/\C[X,Y]_{(f)}$ and then we map $\C(X,Y)\rightarrow\bigoplus_f\C(X,Y))/\C[X,Y]_{(f)}$ which is the next step in an injective resolution, the kernel of this map is clearly $\C[X,Y]$. Finally, the cokernel of this map is injective (as the global dimension of $\C[X,Y]$ is $2$).

Addendum: A systematic way of getting this resolution as well as identifying the last term is to note that the Cousin complex of $\C[X,Y]$ is an injective resolution (Hartshorne: Residues and duality, SLN 20, p. 239) which in degree $p$ is the sum of the injective hulls of the residue fields of points of dimension $p$.

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