Integral versions of the Nakayama and finitistic dimension conjectures
type: Item
instance of: scholarly article
title: Integral versions of the Nakayama and finitistic dimension conjectures
zbMATH Open document ID: 0813.16010
author: Alfred Wiedemann
published in: Journal of Algebra
publication date: 1994-12-19T00:00:00Z
review text: The author proves the (generalized) Nakayama conjecture and the finitistic dimension conjecture for lattices over classical orders in special cases. These cases are somehow the integral analogue of the cases which where considered for artinian algebras by \textit{P. Dräxler} and \textit{D. Happel} [J. Pure Appl. Algebra 78, No. 2, 161-164 (1992; Zbl 0794.16007)]. More precisely, let \(R\) be a complete Dedekind domain with field of fractions \(K\), and let \(\Lambda\) be an \(R\)-order in a separable \(K\)-algebra \(A\). For a left \(\Lambda\)-lattice \(M\), the dual \(\text{Hom}_ R(M,R)\) is denoted by \(M^*\). As for algebras, the generalized Nakayama conjecture is equivalent to the following statement: For each simple \(\Lambda\)-module \(S\), there exists an integer \(i \geq 0\) such that \(\text{Ext}^ i_ \Lambda(\Lambda^*,S) \neq 0\). To avoid technicalities, let me just state a corollary of one of the main results: The generalized Nakayama conjecture holds for \(\Lambda\) unless there exists a proper chain of over-orders \(\Lambda \subset \Lambda_ 1 \subset \Lambda_ 2\) such that \(\Lambda_ 2\) is of infinite lattice type. The author describes a class of orders \(\Lambda\), where the ring of multipliers of the radical is a special Bäckström order, for which the finitistic dimension conjecture holds. He also gives illuminating examples where his results can be applied to prove the finitistic dimension conjecture.
reviewed by: Klaus W. Roggenkamp
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 703225
zbMATH Keywords: finitistic dimension conjecture
zbMATH Keywords: lattice over classical orders
zbMATH Keywords: complete Dedekind domain
zbMATH Keywords: separable \(K\)-algebra
zbMATH Keywords: generalized Nakayama conjecture
zbMATH Keywords: simple \(\Lambda\)-module
zbMATH Keywords: infinite lattice type
zbMATH Keywords: Bäckström order
Wikidata QID: Q122925566
MaRDI profile type: Publication
full work available at URL: jabr.1994.1343
OpenAlex ID: W2080876166
DOI: 10.1006/JABR.1994.1343
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description: scientific article; zbMATH DE number 703225