Invariants of Linkage of modules

TL;DR

This paper explores invariants related to linkage of Cohen-Macaulay modules over Gorenstein local rings, establishing equivalences of vanishing Ext and Tor groups, and revealing symmetries in endomorphism rings.

Contribution

It introduces new invariance results for linked modules, connecting Ext and Tor vanishing conditions, and provides insights into endomorphism ring structures, answering a previously open question.

Findings

Equivalence of vanishing Ext and Tor groups for linked modules.

Symmetry in endomorphism rings of linked modules.

Negative answer to a question by Martsinkovsky and Strooker.

Abstract

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M, N$ be two Cohen-Macaulay \ $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $Ext^i_A(L,M) = 0 $ for all $i \gg 0$ if and only if $Tor^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is Cohen-Macaulay then we show that $Ext^i_A(M, D) = 0 $ for all $i \gg 0$ if and only if $Ext^i_A(D^\dagger, N) = 0$ for all $i \gg 0$, where $D^\dagger = Ext^r_A(D,A)$ and $r = codim \ D$. As a consequence we get that $Ext^i_A(M, M) = 0 $ for all $i \gg 0$ if and only if $Ext^i_A(N, N) = 0$ for all $i \gg 0$. We also show that $End_A(M)/rad \ End_A(M) \cong (End_A(N)/rad \ End_A(N))^{op}$. We also give a negative answer to a question of Martsinkovsky and Strooker.