Gorenstein Homological Dimensions Research Articles

First, Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.

Ascent Properties of Auslander Categories

Abstract. Let R be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over R. We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For example, we establish base change relations between the Auslander categories of the source and target rings of a homomorphism φ: R → S of finite flat dimension.

Finitistic dimension and restricted flat dimension

We investigate the relations between finitistic dimensions and restricted flat dimensions (introduced by Foxby [L.W. Christensen, H.-B. Foxby, A. Frankild, Restricted homological dimensions and Cohen–Macaulayness, J. Algebra 251 (1) (2002) 479–502]). In particular, we show the following result. (1) If T is a selforthogonal left module over a left noetherian ring R with the endomorphism ring A, then rfd ( T A ) ⩽ fdim ( A A ) ⩽ id ( T R ) + rfd ( T A ) . (2) If T R is classical partial tilting, then fdim ( A A ) ⩽ fdim ( R R ) + rfd ( T A ) . (3) If A = A 0 ⊆ A 1 ⊆ ⋯ ⊆ A m = R are Artin algebras with the same identity such that, for each 0 ⩽ i ⩽ m − 1 , rad A i is a right ideal in A i + 1 and rfd ( A i + 1 A i ) < ∞ (e.g., A i + 1 A i is of finite projective dimension, or finite Gorenstein projective dimension, or finite Tor-bound dimension), then fdim ( R R ) < ∞ implies fdim ( A A ) < ∞ . As applications, we disprove Foxby's conjecture [H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004) 167–193] on restricted flat dimensions by providing a counterexample and give a partial answer to a question posed by Mazorchuk [V. Mazorchuk, On finitistic dimension of stratified algebras, arXiv:math.RT/0603179, 6.4].

On Gorenstein projective, injective and flat dimensions—A functorial description with applications

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the noncommutative realm, k-algebras with a dualizing complex.