Linkage of modules and the Serre conditions

Abstract

Abstract. Let R be semiperfect commutative Noetherian ring and C be a semidualizing Rmodule. The connection of the Serre condition (Sn) on a horizontally linked R-module of finite GC -dimension with the vanishing of certain cohomology modules of its linked module is discussed. As a consequence, it is shown that under some conditions Cohen-Macaulayness is preserved under horizontal linkage.

Linkage of finite GC-dimension modules

“…(i) M satisfies S n ; (ii) M is an n-th syzygy module; (iii) M is horizontally linked and Ext i R (λM, K) = 0 for all i, 0 < i < n.Proof. This is an immediate consequence of Corollary 4.2, Example 2.7(iv) and[5, Corollary 2.14]. Let R be a semiperfect ring, M a horizontally linked R-module of finite G Cdimension and λM ∈ A C (e.g.…”

“…(iii)⇒(iv). By [5,Theorem 4.6], r. grade(M, C) ≥ n. Now assume contrarily that G C -dim R (M ) > 0. Hence we obtain the following inequality from Theorem 2.5:…”

“…In [4] and [5], Dibaei and Sadeghi studied the theory of linkage for modules of finite G Cdimension. In this paper, we continue our study and obtain new results in this direction.…”

On reduced G-perfection and horizontal linkage relative to a semidualizing module

“…The following relative notion, suggested in [9], extends this concept by taking into account a semidualizing module. Definition 3.1.…”

“…More specifically, as anticipated in the abstract, one of our main goals is to contribute to the theory mostly by extending, non-trivially, some of the results of Dibaei and Sadeghi [8] to the relative context taking into account the presence of C. Such relative theory, including the property of reduced G C -perfection as well as the relative operations of Auslander transpose and horizontal linkage, was developed by the same authors in [9], [10], also Sadeghi [23]; here it should be mentioned that the idea of relative Auslander transpose is due to Foxby [11]. Along the way, we furthermore improve results of Auslander and Bridger [4] as well as of Martsinkovsky and Strooker [21], and provide examples illustrating properties which, as far as we know, have gone unnoticed in the literature.…”