Explicit injective resolutions of (Laurent) polynomial rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T14:25:49Z http://mathoverflow.net/feeds/question/47817 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47817/explicit-injective-resolutions-of-laurent-polynomial-rings Explicit injective resolutions of (Laurent) polynomial rings Maxime Bourrigan 2010-11-30T17:54:18Z 2010-12-01T09:18:25Z <p>Hi,</p> <p>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 <em>Cohomology of sheaves</em> (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.</p> <p>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. </p> <p>Thanks!</p> http://mathoverflow.net/questions/47817/explicit-injective-resolutions-of-laurent-polynomial-rings/47823#47823 Answer by Torsten Ekedahl for Explicit injective resolutions of (Laurent) polynomial rings Torsten Ekedahl 2010-11-30T19:00:13Z 2010-12-01T09:18:25Z <p>$\newcommand{\C}{\mathbb C}$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 $2$).</p> <p><b>Addendum</b>: 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 $p$.</p>