Injective resolutions of some regular rings
The authors investigate the extent to which the homogeneity known to hold for the minimal injective resolution of a commutative Noetherian local Gorenstein ring carries over to a noncommutative Noetherian Auslander-Gorenstein ring \(A\). In this context, the desired homogeneity is phrased as a purity condition with respect to the canonical dimension function on \(A\)-modules, given by \(\delta(M)=d-\text{grade}(M)\), where \(d=\text{inj.}\dim(A)\). An \(A\)-module is called \(s\)-pure if all its nonzero Noetherian submodules \(N\) satisfy \(\delta(N)=s\), and essentially \(s\)-pure if it contains an essential submodule which is \(s\)-pure. Finally, \(A\) is said to have a pure (essentially pure) injective resolution if the \(i\)-th term of the minimal injective resolution of \(A_A\) is always \((d-i)\)-pure (essentially \((d-i)\)-pure). The authors showed earlier that a pure minimal injective resolution exists when \(A\) is grade-symmetric and satisfies a polynomial identity [Commun. Algebra 26, No. 7, 2159-2180 (1998; Zbl 0910.16005)]. However, even essentially pure resolutions do not exist in general, as shown by examples of Artin and Stafford [in loc. cit.]. In the present paper, it is proved that pure or essentially pure minimal injective resolutions occur for several classes of examples with \(d\leq 4\). For instance, pure resolutions occur for Auslander-Gorenstein domains \(A\) with injective dimension 2 such that every simple \(A\)-module with grade 2 is annihilated by a completely prime normal element, and that essentially pure resolutions occur for the analogous Auslander-Gorenstein domains with injective dimension 3. Essentially pure graded injective resolutions are obtained for the four-dimensional Sklyanin algebra and for the homogenized enveloping algebra of any three-dimensional Lie algebra (over an algebraically closed field). The authors' route to these results involves some work with noncommutative Cohen-Macaulay conditions that is interesting in itself. In particular, they prove that a Noetherian ring of injective dimension 3 which is Cohen-Macaulay with respect to Gelfand-Kirillov dimension must be Auslander-Gorenstein. (Also submitted to MR).
scientific article; zbMATH DE number 1330822