Integrally Closed Ideals of Reduction Number Three

DOI10.1142/S1005386723000263zbMATH Open1516.13017arXiv2105.07186MaRDI QIDQ6098064

Publication date: 9 June 2023

Published in: Algebra Colloquium (Search for Journal in Brave)

Abstract: In a Cohen-Macaulay local ring

(A, m a t h f r a k m)

, we study the Hilbert function of an integrally closed

m a t h f r a k m

-primary ideal

I

whose reduction number is three. With a mild assumption we give an inequality

e l l_{A} (A / I) g e m a t h r m e_{0} (I) - m a t h r m e_{1} (I) + d f r a c m a t h r m e_{2} (I) + e l l_{A} (I^{2} / Q I) 2

, where

m a t h r m e_{i} (I)

denotes the

i

th Hilbert coefficients and

Q

denotes a minimal reduction of

I

. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of

I

is larger than or equal to

d i m A - 1

. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.

Full work available at URL: https://arxiv.org/abs/2105.07186

zbMATH Keywords

Cohen-Macaulay ring Hilbert function associated graded ring Rees algebra Sally module

Mathematics Subject Classification ID

Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) (13H10) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series (13D40) Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics (13A30)