Betti numbers of modules of exponent two over regular local rings
type: Item
instance of: scholarly article
title: Betti numbers of modules of exponent two over regular local rings
zbMATH Open document ID: 0882.13017
author: Shou-Te Chang
published in: Journal of Algebra
publication date: 1998-03-05T00:00:00Z
review text: Let \((R, {\mathfrak m},k)\) be a regular local ring of dimension \(n\) and \(M\) an \(R\)-module of finite length. Horrocks posed two questions: (1) Is the \(i\)-th Betti number of \(M\) at least \(n\choose i\)? (2) Is the rank of the \(i\)-th syzygy of \(M\) at least \({n-1 \choose i-1}\)? Of course the affirmative answer of the second question gives the affirmative answer of the first one and we have the affirmative answer of these questions for modules annihilated by \({\mathfrak m}\). The present paper gives the affirmative answer of these questions for modules annihilated by \({\mathfrak m}^2\). This paper contains a further result. Let \(r,s\) and \(i\) be positive integers. Then there is an integer \(a\geq{n-1 \choose i-1}\), depending only on \(n,i,r\) and \(s\), such that the rank of the \(i\)-th syzygy of \(M\geq a\) for any \(R\)-module \(M\) with exact sequence \(0\to k^s \to M \to k^r \to 0\) and the equality holds for ``almost all'' \(M\).
reviewed by: Takesi Kawasaki
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 13D03
zbMATH DE Number: 1051580
zbMATH Keywords: module of finite length
zbMATH Keywords: regular local ring
zbMATH Keywords: Betti number
zbMATH Keywords: syzygy
MaRDI profile type: Publication
full work available at URL: jabr.1996.7018
OpenAlex ID: W1985317508
cites work: Obstructions to the Existence of Multiplicative Structures on Minimal Free Resolutions
cites work: Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3
cites work: Generic free resolutions and a family of generically perfect ideals
cites work: What makes a complex exact?
cites work: Q4721262
cites work: Q3479217
cites work: Betti numbers of multigraded modules
cites work: Q4026068
cites work: Betti Numbers for Modules of Finite Length
cites work: Betti numbers of almost complete intersections
cites work: Q3686801
cites work: Binomial behavior of Betti numbers for modules of finite length
cites work: Algebraic vector bundles on projective spaces: A problem list
cites work: On the bettinumbers of finite pure and linear resolutions
cites work: Vector Bundles on the Punctured Spectrum of a Local Ring
cites work: The structure of linkage
cites work: Horrocks' question for monomially graded modules
DOI: 10.1006/JABR.1996.7018
description: scientific article; zbMATH DE number 1051580