Approximating modules by modules of finite projective dimension
type: Item
instance of: scholarly article
title: Approximating modules by modules of finite projective dimension
zbMATH Open document ID: 0846.16012
author: W. D. Burgess
author: Birge Zimmermann Huisgen
published in: Journal of Algebra
publication date: 1996-09-25T00:00:00Z
review text: Let \(\Lambda\) be a finite dimensional split left serial algebra over a field \(K\) and let \(P^\infty\) be the full subcategory of \(\Lambda\)-mod consisting of all modules of finite projective dimension. The authors establish the contravariant finiteness of \(P^\infty\) in \(\Lambda\)-mod in the sense of Auslander et al. Each finitely generated \(\Lambda\)-module \(M\) has a right \(P^\infty\)-approximation, i.e. there exists a homomorphism \(f: A\to M\) with \(A \in P^\infty\) such that each homomorphism \(B \to M\) with \(B \in P^\infty\) factors through \(f\). Moreover the authors give an explicit description of the structure of the minimal \(P^\infty\)-approximations \(A_1, \dots, A_n\) of the simple left \(\Lambda\)-modules. By a result of \textit{M. Auslander} and \textit{I. Reiten} [Adv. Math. 86, No. 1, 111-152 (1991; Zbl 0774.16006)] these modules \(A_1, \dots, A_n\) are the building blocks of the modules of finite projective dimension.
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 826741
zbMATH Keywords: finite dimensional split left serial algebras
zbMATH Keywords: modules of finite projective dimension
zbMATH Keywords: contravariant finiteness
zbMATH Keywords: finitely generated modules
zbMATH Keywords: right \(P^ \infty\)-approximations
zbMATH Keywords: minimal \(P^ \infty\)-approximations
zbMATH Keywords: simple left modules
MaRDI profile type: Publication
full work available at URL: jabr.1995.1340
OpenAlex ID: W1983268653
DOI: 10.1006/JABR.1995.1340
description: scientific article; zbMATH DE number 826741