A study of quasi-Gorenstein rings

Abstract

In this paper several quasi-Gorenstein counterparts to some known properties of Gorenstein rings are given. We, furthermore, give an explicit description of the attached prime ideals of certain local cohomology modules.2010 Mathematics Subject Classification. 13H10, 13D45.

A Study of quasi-Gorenstein rings II: Deformation of quasi-Gorenstein property

“…(R p ) for each p ∈ Spec(R/yR). Since both R p and R p /y n R p are quasi-Gorenstein for each p ∈ Spec • (R/yR), we have C p = 0 for p ∈ Spec • (R/y n R) in view of [26,Corollary 2.8] and hence our claim follows. There is a group homomorphism:…”

“…Thus, the local ring R that appears in Main Theorem 1 is not Cohen-Macaulay. In the absence of Cohen-Macaulay condition, various aspects have been studied around the deformation problem in a recent paper [26]. Our second main result is to provide some conditions under which the quasi-Gorenstein condition is preserved under deformation (see Theorem 3.2).…”

Reduction of the Small Cohen–Macaulay Conjecture to excellent unique factorization domains

“…we can conclude that R belongs to the class A . Note also that by [54,Theorem 5.12. (i)], roughly speaking, A is a subclass of generalized Cohen-Macaulay rings.…”

“…for some system of parameters x of R. Note that A contains, properly, the class of Buchsbaum rings (see, [54,Theorem 5.12(ii)]). In order to see why this inclusion is proper, note that in the light of [19], there exists a non-Buchsbaum quasi-Buchsbaum ring R such that there are exactly two non-zero non-top local cohomologies of R which both of them are R/mvector spaces.…”

Reductions towards a characteristic free proof of the Canonical Element Theorem

“…For the details see Remark 2.7 of the arXiv version of [TaTu18]. For another proof of this lemma and its converse assuming only that depth(S) > 0 and dim(S) ≥ 3, see Lemma 2.5 of the arXiv version of [TaTu18].…”