Bound on the a-invariant and reduction numbers of ideals

Details Bound on the a-invariant and reduction numbers of ideals Bound on the a-invariant and reduction numbers of ideals

2004-04-04

arxiv.org/abs/math/0404066v1

Mathematics

Commutative Algebra

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

Scientific paper

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 $H^i_M(R)_n$, 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 $$EG(R)= \sum_{q=0}^{d-1} \binom{d-1}{q} h^q(R)_{1-q}.$$ We prove that the $a$-invariant of $R$ satisfies $$ a(R) \leq e(R)-length(R_1)+(d-1)(length(R_0)-1)+ EG(R).$$ 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.

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