Bound on the a-invariant and reduction numbers of ideals

Abstract

Let $R$ be a $d$-dimensional standard graded ring over an Artin local ring. Let $M$ be the unique maximal homogeneous ideal of $R.$ Let $h^i(R{)}_n$ denote the length of $H_M^i(R{)}_n$, i.e. the nth graded component of the ith local cohomology module of R with respect to M. Define the Eisenbud-Goto invariant of $R$ to be the number $$EG(R)=\sum _{q=0}^{d-1}\left(\genfrac{}{}{0ex}{}{d-1}{q}\right)h^q(R{)}_{1-q}.$$ We prove that the $a$-invariant of $R$ satisfies $$a(R)\le e(R)-length(R_1)+(d-1)(length(R_0)-1)+EG(R).$$ Using this bound we get upper bounds for the reduction number of an $m$-primary ideal of a Cohen-Macaulay local ring $(R,m)$ whose associated graded ring $G(m)$ has almost maximal depth.

8 pages. to appear in Journal of algebra 274(2004) 594-601