Bound on the a-invariant and reduction numbers of ideals

2004/04/04 by Clare D’Cruz, Clare D'Cruz, Vijay Kodiyalam +5
Computer Science · Mathematics · #13D45 14B15 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras #math.AC #msc:13D45 #msc:14B15

paper · pdf · doi:10.48550/arxiv.math/0404066

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

arxiv created 2004/04/04 · openalex publication_date 2004/04/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 hⁱ(R)_n denote the length of Hⁱ_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)= ∑_q=0^d-1 \binomd-1q h^q(R)_1-q. We prove that the a-invariant of R satisfies a(R) ≤ e(R)-length(R₁)+(d-1)(length(R₀)-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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