Auslander Modules and Quasi-Homogeneity of Local Rings

Abstract

§0. IntroductionLet (R,m,k) be a normal local domain of dimension two and assume that k is an algebraically closed field of characteristic zero. Let us denote K R the canonical module of R. Then from the definition of K R , there are isomorphisms:which involves the unique existence of the nonsplit exact sequence:The J?-module A which appears in the sequence is also uniquely determined except for isomorphisms, and is called the Auslander module of R. (For more details, see [13; Chapter 11].) In this paper, it is of our main interest how the Auslander module is related to the quasi-homogeneity of a local ring. Here recall that R is said to be quasi-homogeneous if it is the completion of a positively graded ring, or equivalently if it has an Euler derivation, i.e. â -derivation A with the property that for some system of generators X l9 X 2 , • • •, X n of m and positive integers d l ,d 2 , --,d n . Now to fix our attention onto a ring of hypersurface singularity, we put R = P/fP where P = k [[X, F,Z]] is the formal power series ring and