The Poincaré series of every finitely generated module over a codimension four almost complete intersection is a rational function
type: Item
instance of: scholarly article
title: The Poincaré series of every finitely generated module over a codimension four almost complete intersection is a rational function
zbMATH Open document ID: 0812.13011
DOI: 10.1016/0022-4049(94)90062-0
author: Andrew Kustin
author: Susan M. Palmer Slattery
published in: Journal of Pure and Applied Algebra
publication date: 1994-12-08T00:00:00Z
review text: The authors prove the following main results. Assume that \(A\) is a quotient of a regular local ring \((R,m,k)\) which is an almost complete intersection of codimension 4 such that 2 is a unit in \(A\). Then the following statements hold true: (i) The Poincaré series \(P^ M_ A (z) = \sum^ \infty_{i=0} \dim \text{Tor}^ A_ i (k,M) \cdot z^ i\) is a rational function for all finitely generated \(A\)-modules \(M\). (ii) If \(R\) contains \(\mathbb{Q}\) then the cotangent modules \(T_ i (A/R,A)\) vanish for \(i\) large if and only if \(A\) is a complete intersection. (iii) For all finitely generated \(A\)-modules \(M\) whose Betti numbers are bounded, the minimal resolution of \(M\) is eventually periodic of period \(\leq 2\).
reviewed by: Lucian Bădescu
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 681808
zbMATH Keywords: regular local ring
zbMATH Keywords: almost complete intersection
zbMATH Keywords: Poincaré series
zbMATH Keywords: Betti numbers
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90062-0
OpenAlex ID: W2019812090
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description: scientific article; zbMATH DE number 681808