Bounds on the a-invariant and reduction numbers of ideals

Abstract

Let R be a d-dimensional standard graded ring over an Artinian local ring. Let M be the unique maximal homogeneous ideal of R. Let h i ( R) n denote the length of the nth graded component of the local cohomology module H i M (R) . Define the Eisenbud–Goto invariant EG( R) of R to be the number ∑ q=0 d−1 d−1 q h q M (R) 1−q. We prove that the a-invariant a( R) of the top local cohomology module H M d(R) satisfies the inequality: a( R)⩽ e( R)−ℓ( R 1)+( d−1)(ℓ( R 0)−1)+ EG( R). This bound is used to get upper bounds for the reduction number of an m -primary ideal I of a Cohen–Macaulay local ring (R, m) , when the associated graded ring of I has depth at least d−1.