A function on the the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to Liason theory

TL;DR

This paper introduces a new function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over Gorenstein rings and applies it to Liason theory, revealing infinite classes in certain cases and finiteness in others.

Contribution

It defines a function on the stable category of MCM modules that interacts well with exact triangles and applies this to derive new results in Liason theory.

Findings

Infinite even liason classes of powers of the maximal ideal in non-regular Gorenstein rings of dimension at least 2.

Finiteness of certain codimension 2 CM-ideals within a bounded multiplicity in complete equi-characteristic simple singularities.

The introduced function helps distinguish liason classes and analyze their structure.

Abstract

Let $(A,\m)$ be a Gorenstein local ring of dimension $d \geq 1$. Let $\CMS(A)$ be the stable category of maximal \CM \ $A$-modules and let $\ICMS(A)$ denote the set of isomorphism classes in $\CMS(A)$. We define a function $\xi \colon \ICMS(A) \rt \ZZ$ which behaves well with respect to exact triangles in $\CMS(A)$. We then apply this to (Gorenstein) liason theory. We prove that if $\dim A \geq 2$ and $A$ is not regular then the even liason classes of $\m^n; n\geq 1$ is an infinite set. We also prove that if $A$ is an complete equi-characteristic simple singularity with $A/\m$ uncountable then for each $m \geq 1$ the set $\mathcal{C}_m = \{I \mid I \ \text{is a codim 2 CM-ideal with} \ e_0(A/I) \leq m \}$ is contained in finitely many even liason classes $L_1,\ldots,L_r$ (here $r$ may depend on $m$).