Special homological dimensions and Intersection Theorem

Let (R,\fm) be commutative Noetherian local ring. It is shown that $R$ is Cohen--Macaulay ring if there exists a Cohen--Macaulay finite (i.e. finitely generated) $R$--module with finite upper Gorenstein dimension. In addition, we show that, in the Intersection Theorem, projective dimension can be replaced by quasi--projective dimension.Comment: 10 page

Stability of Gorenstein Categories

We show that an iteration of the procedure used to define the Gorenstein projective modules over a commutative ring $R$ yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective $R$-modules G=...\xra{\partial^G_2}G_1\xra{\partial^G_1}G_0\xra{\partial^G_0} ... such that the complexes \Hom_R(G,H) and \Hom_R(H,G) are exact for each Gorenstein projective $R$-module $H$, the module \coker(\partial^G_1) is Gorenstein projective. The proof of this result hinges upon our analysis of Gorenstein subcategories of abelian categories.Comment: 21 pages, uses XY-pic. Version 2 contains corrected proofs of Lemma 2.1 and Theorem 4.

Cohen-Macaulay homological dimensions

We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.Comment: To appear in Mathematica Scandinavic