A FUNCTION ON THE SET OF ISOMORPHISM CLASSES IN THE STABLE CATEGORY OF MAXIMAL COHEN-MACAULAY MODULES OVER A GORENSTEIN RING: WITH APPLICATIONS TO LIAISON THEORY

Let (A, m) be a Gorenstein local ring of dimension d >= 1. Let (CM) under bar (A) be the stable category of maximal Cohen-Macaulay A-modules and let (ICM) under bar (A) denote the set of isomorphism classes in (CM) under bar (A). We define a function xi:(ICM) under bar (A) -> Z which behaves well with respect to exact triangles in CM(A). We then apply this to (Gorenstein) liaison theory. We prove that if dim A >= 2 and A is not regular then the even liaison classes of {m(n) vertical bar n >= 1} is an infinite set. We also prove that if A is Henselian with finite representation type with Alm uncountable then for each m >= 1 the set l(m) = {I vertical bar I is a codim 2 CM-ideal with eo (A/I) <= m} is contained in finitely many even liaison classes L-1, ..., L-r (here r may depend on m)