Shellability of simplicial complexes arising in representation theory
type: Item
instance of: scholarly article
title: Shellability of simplicial complexes arising in representation theory
zbMATH Open document ID: 0932.16006
author: Luise Unger
published in: Advances in Mathematics
publication date: 1999-12-15T00:00:00Z
review text: Let \(k\) be an algebraically closed field and let \(A\) be a finite-dimensional \(k\)-algebra. Recall that an \(A\)-module \(T\) is called tilting provided: (a) the projective dimension of \(T\) is finite; (b) \(\text{Ext}^i_A(T,T)=0\) for each \(i>0\); and (c) there exists an exact sequence \(0\to A\to T_0\to T_1\to\cdots\to T_r\to 0\), with \(T_i\in\text{add }T\). Tilting theory is nowadays an important tool to understand the category of \(A\)-modules. As observed by the author, C. Ringel has noticed that the set of tilting \(A\)-modules forms a simplicial complex \(\Sigma\) and suggested its study. One of the aims of the paper under review is to give a uniform approach to this study made mainly by Riedtmann, Schofield and Unger. More than that, the author has deeepened the results and has related them with some other lines of research in representation theory.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 1334588
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: categories of modules
zbMATH Keywords: tilting modules
zbMATH Keywords: simplicial complexes
reviewed by: Flávio U. Coelho
MaRDI profile type: Publication
full work available at URL: aima.1998.1812
OpenAlex ID: W2078382762
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DOI: 10.1006/AIMA.1998.1812
description: scientific article; zbMATH DE number 1334588