Complete intersection augmented algebras
type: Item
instance of: scholarly article
title: Complete intersection augmented algebras
zbMATH Open document ID: 0864.13009
DOI: 10.1007/BF01445245
published in: Mathematische Annalen
publication date: 1997-06-23T00:00:00Z
full work available at URL: https://eudml.org/doc/165448
review text: In his work in Ann. Math., II. Ser. 141, No. 3, 443-551 (1995; Zbl 0823.11029), \textit{A. Wiles} used a certain criterion in order to prove that a particular Hecke algebra is a one-dimensional complete intersection. This follows by counting the length of the co-normal module of an augmentation ideal that is torsion. The main aim of the paper under review is to give a higher-dimensional generalization of this criterion. To this end let \((R,{\mathfrak m})\) be a local Gorenstein ring and \(A\) an augmented \(R\)-algebra with augmentation ideal \(N_A:= \ker \pi_A\), where \(\pi_A:A\to R\) denotes the augmentation map. Suppose that \(\dim A/{\mathfrak m} A=0\) and that \(N_A/N^2_A\) is torsion as \(R\)-module. Then \(A\) is locally a complete intersection if and only if \(A_{M_A}\), \(M_A={\mathfrak m} +N_A\), is a Gorenstein ring and the congruence ideal \(\eta_A: =\pi_A(0:N_A)\) is a principal ideal with \(F_0(N_A/N^2_A) =\eta_A\), where \(F_0\) denotes the 0-th Fitting ideal. In the particular case of dim \(R=1\) there are a few corollaries, including A. Wiles' result. The main technical tools for the author's proof are some basic facts about linkage and the André-Quillen homology group for showing that a certain ideal is syzygetic.
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 931995
zbMATH Keywords: linkage
zbMATH Keywords: augmented algebra
zbMATH Keywords: Hecke algebra
zbMATH Keywords: complete intersection
zbMATH Keywords: local Gorenstein ring
author: Santiago Zarzuela
MaRDI profile type: Publication
cites work: Algèbres graduees associees et algèbres symétriques plates
cites work: Syzygetic ideals, regular sequences, and a question of Simis
cites work: Q4745961
cites work: Q4858022
cites work: Local Euler characteristics
cites work: Liaison des variétés algébriques. I
cites work: On some five-term exact sequences in homology
cites work: Notes on liaison and duality
cites work: On the symmetric and Rees algebras of an ideal
cites work: Q4293985
cites work: Modular elliptic curves and Fermat's Last Theorem
OpenAlex ID: W2071922491
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description: scientific article; zbMATH DE number 931995