Global dimension two orders are quasi-hereditary
type: Item
instance of: scholarly article
title: Global dimension two orders are quasi-hereditary
zbMATH Open document ID: 0681.16004
DOI: 10.1007/BF02568479
author: Alfred Wiedemann
author: Steffen Koenig
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155456
review text: Let R be a complete discrete valuation ring with field of fractions K. Let \(\Lambda\) be a basic R-order in a separable K-algebra A, and let \(f_ 1,...,f_ s\) be a full set of primitive central idempotents of A. \(\Lambda\) is said to be quasi-hereditary if it contains primitive idempotents \(e_ 1,...,e_ s\) such that 1) \(f_ ie_ i=e_ i\) for each i; 2) gl.dim \(e_ i\Lambda e_ i<\infty\) for each i; and 3) For \(e=\sum e_ i\), the ring \(\Lambda\) /\(\Lambda\) \(e\Lambda\) is quasi- hereditary in the sense of \textit{E. Cline}, \textit{B. Parshall}, and \textit{L. Scott} [J. Reine Angew. Math. 391, 85-99 (1988; Zbl 0657.18005)]. The authors prove analogs for this situation of some results of \textit{V. Dlab} and \textit{C. M. Ringel} [Ill. J. Math. 33, 280-291 (1989; Zbl 0666.16014)]. To wit, if gl.dim \(\Lambda\leq 2\) then \(\Lambda\) is quasi- hereditary, and \(\Lambda\) is hereditary if and only if every chain of full, two-sided, idempotent ideals can be refined to a so-called heredity chain for \(\Lambda\). \{Reviewer's remark: The use of the noun ``heredity'' as an adjective is very unpleasant. If it is not already too late, a better term should be chosen.\}
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16Gxx
zbMATH DE Number: 4114816
zbMATH Keywords: complete discrete valuation ring
zbMATH Keywords: basic R-order
zbMATH Keywords: separable K-algebra
zbMATH Keywords: primitive central idempotents
zbMATH Keywords: quasi-hereditary
zbMATH Keywords: idempotent ideals
zbMATH Keywords: heredity chain
reviewed by: William H. Gustafson
MaRDI profile type: Publication
cites work: A characterization of orders of finite lattice type
cites work: Algebraic stratification in representation categories
cites work: Q3804882
cites work: Q3922829
cites work: Quasi-hereditary algebras
cites work: On hereditary orders
cites work: Quasi-hereditary orders
cites work: Q3808313
OpenAlex ID: W2061600007
description: scientific article; zbMATH DE number 4114816