Local Artinian rings and the Fröberg relation
type: Item
instance of: scholarly article
title: Local Artinian rings and the Fröberg relation
zbMATH Open document ID: 0877.13017
author: Robert W. Fitzgerald
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-09-04T00:00:00Z
full work available at URL: CONT26-4.html
review text: Let \(R\) be a commutative Artinian local ring, with maximal ideal \({\mathfrak m}\), and residue field \(k\). Define the Poincaré series \(P_R(t)= \sum_{i\geq 0}\dim(\text{Tor}_i^R(k,k))t^i\) and the Hilbert series \(H_R(t)= \sum_{i\geq 0}\dim ({\mathfrak m}^i/{\mathfrak m}^{i+1})t^i\). \(R\) is called a Fröberg ring if \(P_R(t)= H_R(-t)^{-1}\). The main result is that if \({\mathfrak m}^3=0\) and \({\mathfrak m}\cdot\text{ann }x={\mathfrak m}^2\) for all \(x\in{\mathfrak m}/{\mathfrak m}^2\), then \(R\) is a Fröberg ring. Some partial converses are given. This is motivated by a preliminary result showing that certain Witt rings are Fröberg rings.
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13E10
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 1036048
zbMATH Keywords: Artinian local ring
zbMATH Keywords: Poincaré series
zbMATH Keywords: Hilbert series
zbMATH Keywords: Fröberg ring
zbMATH Keywords: Witt rings
reviewed by: William H. Gustafson
MaRDI profile type: Publication
OpenAlex ID: W2032058806
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DOI: 10.1216/RMJM/1181071992
description: scientific article; zbMATH DE number 1036048