On the derived categories and quasitilted algebras.
type: Item
instance of: scholarly article
title: On the derived categories and quasitilted algebras.
zbMATH Open document ID: 1184.16009
author: Cecilia Tosar
author: Flávio U. Coelho
published in: Algebras and Representation Theory
publication date: 2009-10-21T00:00:00Z
review text: The authors prove the following results and derive from them some corollaries about quasi-tilted algebras: Theorem. Let \(A\) be an algebra. If every indecomposable complex in the bounded derived category of \(A\)-modules has length (the number of nonzero components) at most \(n\) for some positive integer \(n\), then the finitistic dimension of \(A\) is at most \(n\). Theorem. A gentle algebra over an algebraically closed field is quasi-tilted if and only if each indecomposable complex in the bounded derived category of \(A\)-modules has length at most two. Theorem. For an algebra \(A\) the following conditions are equivalent: (i) \(A\) is quasi-tilted. (ii) Every complex in the bounded derived category of \(A\)-modules is isomorphic to a complex, consisting of a module \(X\) such that from the existence of a finite path of nonzero morphisms in \(\text{ind\,}A\) from \(Y\) to \(X\) it follows that the projective dimension of \(Y\) is at most \(1\). (iii) Every complex in the bounded derived category of \(A\)-modules is isomorphic to a complex, consisting of a module \(Y\) such that from the existence of a finite path of nonzero morphisms in \(\text{ind\,}A\) from \(Y\) to \(X\) it follows that the injective dimension of \(X\) is at most \(1\).
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 16E35
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 5619085
zbMATH Keywords: derived categories
zbMATH Keywords: quasi-tilted algebras
zbMATH Keywords: modules
zbMATH Keywords: gentle algebras
zbMATH Keywords: finitistic dimension
MaRDI profile type: Publication
full work available at URL: s10468-008-9105-6
OpenAlex ID: W1976053782
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DOI: 10.1007/S10468-008-9105-6
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description: scientific article; zbMATH DE number 5619085