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RELATIVE HILBERT CO-EFFICIENTS

Part of: Commutative algebra: Homological methods General commutative ring theory

Published online by Cambridge University Press:  01 March 2017

AMIR MAFI ,
TONY J. PUTHENPURAKAL ,
RAKESH B. T. REDDY  and
HERO SAREMI
AMIR MAFI
Affiliation:
Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran e-mail: a_mafi@ipm.ir
TONY J. PUTHENPURAKAL
Affiliation:
Department of Mathematics, IIT Bombay, Powai, Mumbai 400 076, India e-mails: tputhen@math.iitb.ac.in, raki.hcu@gmail.com
RAKESH B. T. REDDY
Affiliation:
Department of Mathematics, IIT Bombay, Powai, Mumbai 400 076, India e-mails: tputhen@math.iitb.ac.in, raki.hcu@gmail.com
HERO SAREMI
Affiliation:
Department of Mathematics, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran e-mail: hero.saremi@gmail.com
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Abstract

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Let (A, ${\mathfrak{m}$) be a Cohen–Macaulay local ring of dimension d and let I ⊆ J be two ${\mathfrak{m}$-primary ideals with I a reduction of J. For i = 0,. . .,d, let eiJ(A) (eiI(A)) be the ith Hilbert coefficient of J (I), respectively. We call the number ci(I, J) = eiJ(A) − eiI(A) the ith relative Hilbert coefficient of J with respect to I. If GI(A) is Cohen–Macaulay, then ci(I, J) satisfy various constraints. We also show that vanishing of some ci(I, J) has strong implications on depth GJn(A) for n ≫ 0.

MSC classification

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Secondary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13D07: Homological functors on modules (Tor, Ext, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 729 - 741
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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