Almost split sequences over \({\mathfrak p}\)-adic rings
type: Item
instance of: scholarly article
title: Almost split sequences over \({\mathfrak p}\)-adic rings
zbMATH Open document ID: 0629.16005
DOI: 10.1007/BF01160208
author: Maria Teresa A. D. Nogueira
author: James A. Green
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173949
review text: Let R be a complete discrete rank one valuation ring with quotient field K, and let \(\Lambda\) be an R-order in a finite-dimensional semisimple K- algebra. \textit{M. Auslander} [in Ring Theory II, Proc. 2nd Okla. Conf. 1975, 1-44 (1977; Zbl 0361.16008)] and independently \textit{K. W. Roggenkamp} and \textit{J. W. Schmidt} [Commun. Algebra 4, 893-917 (1976; Zbl 0361.16007)] proved that almost split sequences exist in the category \(mod^ 0\Lambda\) of \(\Lambda\)-lattices. More precisely, if \(M\in mod^ 0\Lambda\) is indecomposable and not projective, these authors show that there is an almost split sequence \({\mathfrak S}: 0\to M^ +\to E\to M\to 0\) in \(mod^ 0\Lambda\), obtainable as follows. Let \(P\to ^{p_ 0}M\to 0\) be a projective cover of M in \(mod^ 0\Lambda\) and let Ṉ: mod\({}^ 0\Lambda \to mod^ 0\Lambda\) be the `Nakayama functor' (ṈX\(=Hom_ R(Hom_{\Lambda}(X,\Lambda),R)\), for \(X\in mod^ 0\Lambda)\). Since Ṉ is covariant and right exact there is an exact sequence \({\mathfrak R}: 0\to M^ +\to \underline NP\to \underline NM\to 0\) in \(mod^ 0\Lambda\), where \(M^ +:=Ker \underline Np_ 0\). The crucial fact is that \(Ext^ 1_{\Lambda}(M,M^ +)\), regarded as \((End_{\Lambda}(M),End_{\Lambda}(M))\)-bimodule, has simple socle: we then take \({\mathfrak S}\) to be the short exact sequence corresponding to (any) nonzero element of this socle. This paper begins with a short proof, based on ideas of Roggenkamp, of Auslander's natural isomorphism \(Ext^ 1_{\Lambda}(X,M^ +)\cong Hom_ R((M,X)/P(M,X),K/R)\) (here \((M,X):=Hom_{\Lambda}(M,X)\), and P(M,X) is the set of all \(f\in (M,X)\) which can be factored through a projective object of \(mod^ 0\Lambda)\). This isomorphism shows that \(Ext^ 1_{\Lambda}(M,M^ +)\) has simple socle; it is implicit in Auslander [loc. cit.], but there does not seem to be an explicit direct proof in the literature. In the present paper it is used to give a criterion for the short exact sequence \({\mathfrak R}\theta\) obtained from \({…
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16Exx
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 4023500
zbMATH Keywords: R-order in a finite-dimensional semisimple K-algebra
zbMATH Keywords: almost split sequences
zbMATH Keywords: \(\Lambda \)-lattices
zbMATH Keywords: projective cover
zbMATH Keywords: Nakayama functor
zbMATH Keywords: simple socle
zbMATH Keywords: pullback
zbMATH Keywords: symmetric Gorenstein order
zbMATH Keywords: minimal projective presentation
zbMATH Keywords: trace
zbMATH Keywords: radical
MaRDI profile type: Publication
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OpenAlex ID: W2080945719
description: scientific article; zbMATH DE number 4023500