We prove that if a positively-graded ring R is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme Tails(R) is a Gorenstein category in the sense of [10]. Moreover, under this condition, a (right) recollement relating Gorensteininjective sheaves in Tails(R) and (graded) Gorenstein-injective R-modules is given.

In this paper, we prove that the global Gorenstein projective dimension of a ring R is equal to the global Gorenstein injective dimension of R, and that the global Gorenstein flat dimension of R is smaller than the common value of the terms of this equality.

Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F (I) of I as reflected in their defining ideals as homomorphic images of polynomial rings over R/I and R/m respectively. In case all the higher conormal modules of I are free over R/I , we observe that: (i) G(I) is C...

Rings of invariants can have nice homological properties even if they do not have finite global dimension. Watanabe’s Theorem [W] gives conditions when the fixed subring of a commutative ring under the action of a finite group is a Gorenstein ring. The Gorenstein condition was extended to noncommutative rings by a condition explored by Idun Reiten in the 1970s, called k-Gorenstein in [FGR]. Thi...

LetR be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed that the equality limn→∞ `(F n R(M)) pnd = `(M) holds when the ring R is a complete intersection or a Gorenstein ring of dimension at most 3. We construct a module over a Gorenstein ring R of dimension five for wh...

Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A ⋉M is a Gorenstein Differential Graded Algebra. As a corollary follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra. Let A be a noetherian local ...

Zaks (1969) proved that the answer is affirmative for a left and right noetherian ring if both dimensions are finite. Such rings are called Gorenstein. For a positive integer k, Auslander and Reiten (1994) initiated the study of k-Gorenstein algebras, which has stimulated several investigations. They showed that the answer to the question above is positive in case is an artin -Gorenstein algebr...

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 rin...

A main ingredient for Kustin–Miller unprojection, as developed in [PR], is the module HomR(I, ωR), where R is a local Gorenstein ring and I a codimension one ideal with R/I Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of [CFHR]. The second and the third are about Tom and Jerry, two famili...

In this note, we introduce the notion of Gorenstein algebras. Let R be a commutative Gorenstein ring and A a noetherian R-algebra. We call A a Gorenstein R-algebra if A has Gorenstein dimension zero as an R-module (see [2]), add(D(AA)) = PA, where D = HomR(−, R), and Ap is projective as an Rpmodule for all p ∈ Spec R with dim Rp < dim R. Note that if dim R = ∞ then a Gorenstein R-algebra A is p...