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 hⁱ(R)_n denote the length of the nth graded component of the local cohomology module Hⁱ_M(R). Define the Eisenbud–Goto invariant EG(R) of R to be the number Σ_q=0^d-1d-1qh^q_M(R)_1-q. We prove that the a-invariant a(R) of the top local cohomology module H_M^d satisfies the inequality: a(R)⩽e(R)−ℓ(R1)+(d−1)(ℓ(R₀)−1)+EG(R). This bound is used to get upper bounds for the reduction number of an -primary ideal I of a Cohen–Macaulay local ring (R,m), when the associated graded ring of I has depth at least d−1.