Bounds on the a-invariant and reduction numbers of ideals

Let R be a d-dimensional standard graded ring over an Artinian local ring. Let 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 . Define the Eisenbud–Goto invariant EG(R) of R to be the number We prove that the a-invariant a(R) of the top local cohomology module 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 , when the associated graded ring of I has depth at least d−1.