Binomial behavior of Betti numbers for modules of finite length
type: Item
instance of: scholarly article
title: Binomial behavior of Betti numbers for modules of finite length
zbMATH Open document ID: 0653.13009
author: Phillip Griffith
author: E. Graham Evans
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let (R,m,k) be a regular local ring of dimension n, M an R-module of finite length, then dim k(Tor\({}\) \(R_ i(k,M))\equiv \beta_ i(M)\) are called the Betti numbers of M, where \(Tor\) \(R_ i(k,M)\) is the i-th syzygy module in a minimal free resolution of M. In the first section of this paper the author establishes fairly general lower bounds on the \(\beta_ i(M)\) if R is isomorphic to the power series ring \(kx_ 1,...,x_ n\). In the second section the following result is obtained: If M is an R-module of monomial type and of finite length, then \(\beta_ j(M)\geq \left( \begin{matrix} \dim (R)\\ j\end{matrix} \right)\) for each j.
reviewed by: Tong Wenting
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 4065136
zbMATH Keywords: module of finite length
zbMATH Keywords: Betti numbers
zbMATH Keywords: syzygy module
zbMATH Keywords: power series ring
MaRDI profile type: Publication
full work available at URL: pjm.1988.133.267
OpenAlex ID: W2092971872
DOI: 10.2140/PJM.1988.133.267
description: scientific article; zbMATH DE number 4065136