Liaison classes of modules

Abstract

We propose a concept of module liaison that extends Gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative Gorenstein ring. Analyzing the resulting equivalence classes we show that several results known for Gorenstein liaison are still true in the more general case of module liaison. In particular, we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules. As a consequence, we show that the intermediate cohomology modules and properties like being perfect, Cohen-Macaulay, Buchsbaum, or surjective-Buchsbaum are preserved in even module liaison classes. Furthermore, we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one.

Linkage and Intermediate C-Gorenstein Dimensions

“…This is completely expected as this linkage was constructed to emulate complete intersection ideal linkage. We also reiterate the more general notion of module linkage, due to Nagel (see [28]).…”

“…The linkage introduced by Nagel [28] generalized Gorenstein linkage for ideals even further. This is accomplished by using modules that play a role that is analogous to that played by Gorenstein ideals.…”

“…In Section 4, we meld together the techniques and ideas from [10], [27], and [28] to connect together the theory of module linkage with these homological dimensions and Serre-like conditions. As was stated above, Nagel generalized module linkage by allowing non-projective modules to present M. In particular, he defines module linkage using quasi-Gorenstein modules, from which we use C-quasi-Gorenstein modules.…”

An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain

“…The focus of [7] is on the interplay between the functors Ext i R (−, R) and the modules. In [8] quasi-Gorenstein modules are defined to facilitate a generalization of ideal linkage to module linkage. These two papers [7,8] provided the inspiration for this paper.…”

Linkage of modules with respect to a semidualizing module

“…The classical linkage theory has been extended to modules by Martin [19], Yoshino and Isogawa [32], Martsinkovsky and Strooker [21], and by Nagel [23], in different ways. Based on these generalizations, several works have been done on studying the linkage theory in the context of modules; see for example [7], [8], [9], [17], [26] and [4].…”