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

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 $m$).