Notes on linkage of modules

Abstract

Let R be a Cohen-Macaulay local ring. It is shown that under some mild conditions, the Cohen-Macaulayness property is preserved under linkage. We also study the connection of (S n ) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module. INTRODUCTIONThe theory of linkage for subschemes of projective space goes back more than a century in some sense, but the modern study was introduced by Peskine and Szpiro [27] in 1974. Recall that two ideals I and J in a Cohen-Macaulay local ring R are said to be linked if there is a regular sequence α in their intersection such that I = (α : J) and J = (α : I). The first main result in the theory of linkage, due to Peskine and Szpiro, indicates that the Cohen-Macaulayness property is preserved under linkage over Gorenstein rings. They also give a counterexample to show that the above result is no longer true if the base ring is Cohen-Macaulay but not Gorenstein. On the other hand, there are interesting extensions of the Peskine-Szpiro Theorem to the case that R is Cohen-Macaulay and one of the ideals I or J is strongly Cohen-Macaulay (i.e. Koszul homology modules are Cohen-Macaulay) [15] or satisfies the sliding depth condition (on Koszul homology) [14]. In a different direction, Schenzel generalized the Peskine-Szpiro Theorem to the vanishing of certain local cohomology modules. More precisely, for linked ideals I and J over a Gorenstein local ring R with maximal ideal m, the Serre condition (S n ) for R/I is equivalent to the vanishing of the local cohomology modules H i m (R/J) = 0 for all i, dim(R/J) − n < i < dim(R/J) [32, Theorem 4.1]. A few authors advanced the theory to the setting of linkage of modules in different ways, for instance Martin [19], Yoshino and Isogawa [35], Martsinkovsky and Strooker [20], and Nagel [24]. Based on these generalizations, several works have been done on studying the linkage theory in the context of modules; see for example [6]-[10], [16], [26], [28] and [29]. In this paper, we are interested in linkage of modules in the sense of [20]. Martsinkovsky and Strooker generalized the notion of linkage for modules by using the composition of two functors: transpose and syzygy. They showed that ideals a and b are linked by zero ideal if and only if R/a ∼ = λ (R/b) and R/b ∼ = λ (R/a), where λ := ΩTr.Recall that the (S k ) locus of M, denoted by S k (M), is the subset of Spec R consisting of all prime ideals p of R such that M p satisfies the Serre condition (S k ). The Gorenstein locus of R, denoted by Gor(R), is the subset of Spec R consisting of all prime ideals p of R such that R p is a Gorenstein local ring. In the fisrt part of this paper, we study the connection of (S n ) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module. As a consequence, we obtain the following result (see also Theorem 3.3 for a more general case).2010 Mathematics Subject Classification. 13C40, 13D45, 13D05, 13C14.

Linkage of modules by reflexive morphisms

“…An R-module M is generalized Cohen-Macaulay if and only if M p is a Cohen-Macaulay R p -module for all p ∈ Spec(R) \ {m} (see [65,Lemmas 1.2,1.4]). Therefore the following result may be seen as a generalization of the Schenzel's result [62,Corollary 3.3] for Gorenstein liaison of ideals as well as the results of Martsinkovsky and Strooker [41,Theorem 11], Nagel [50,Corollary 6.1(b)] and Sadeghi [61,Corollary 5.4] for their smaller module liaison classes.…”

“…Schenzel proved that Recall that an R-module M is called t-torsionfree with respect to K provided that Ext i R (Tr K M, K) = 0 for all i, 1 ≤ i ≤ t. For an ideal a of R, we denote by V(a) the set of all prime ideals of R containing a. The following is a generalization of [61,Theorem 5.1]. Lemma 4.18.…”