The derived category of a hereditary order
type: Item
instance of: scholarly article
title: The derived category of a hereditary order
zbMATH Open document ID: 0832.16012
DOI: 10.1007/BF01195091
author: Dieter Happel
author: Klaus W. Roggenkamp
published in: Archiv der Mathematik
publication date: 1996-02-18T00:00:00Z
review text: We give a description of the Auslander-Reiten quiver of the derived category of bounded complexes of lattices for a connected hereditary order \(\Lambda\). Let us assume that \(\Lambda\) has \(n\) non-isomorphic indecomposable lattices. The components of the Auslander-Reiten quiver are of two types: \(\mathbb{Z} A_\infty\) modulo the cyclic group of order \(n\), for each integer \(i\) we have such a component \(T_n (i)\); these correspond to indecomposable complexes of length two with torsion homology. Moreover, there are components of type \(N/\langle n\rangle\); i.e. an oriented sphere with \(n\) vertices. For each integer \(i\) we have such a component \(S_n (i)\). These components correspond to the complexes of lattices, concentrated in one degree. Between the various components there are only morphisms \(T_n (i) \to S_n(i + 1)\) and \(S_n(i) \to T_n(i)\). For each \(i\) the objects in \(S_n(i) \cup T_n (i)\) are in bijection to the indecomposable \(\Lambda\)-modules. Moreover, in the derived category they are isomorphic to the category of \(\Lambda\)-modules. We also show that for the Abelian category of the \(R\)-torsion \(\Lambda\)- modules \(\mathcal T\), the derived category \({\mathcal D}^b({\mathcal T})\) has Auslander-Reiten triangles, and that the Auslander-Reiten quivers of both derived categories coincide. Moreover, we present an example of S. König, which shows that a hereditary order \(\Lambda\) is derived equivalent to a ring \(\Gamma\), which is not an order. This example has the property that 1. Though \(\Lambda\) and \(\Gamma\) are derived equivalent, there does not exist a Rickard-complex, realizing the derived equivalence, 2. If \(F\) is the residue field of \(R\), the \(F\)-algebras \(F\otimes_R\Lambda\) and \(F\otimes_R\Gamma\) are not derived equivalent.
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 815706
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: derived category of bounded complexes of lattices
zbMATH Keywords: connected hereditary orders
zbMATH Keywords: indecomposable lattices
zbMATH Keywords: components
zbMATH Keywords: Auslander-Reiten triangles
zbMATH Keywords: derived equivalences
MaRDI profile type: Publication
cites work: Hereditary orders and artin algebras of finite representation type
cites work: Q3994341
cites work: Q4061092
cites work: Auslander-Reiten triangles in derived categories
full work available at URL: https://doi.org/10.1007/bf01195091
OpenAlex ID: W2063178858
description: scientific article; zbMATH DE number 815706