A short proof for Auslander's defect formula.
type: Item
instance of: scholarly article
title: A short proof for Auslander's defect formula.
zbMATH Open document ID: 1035.18007
DOI: 10.1016/S0024-3795(02)00481-0
author: Henning Krause
published in: Linear Algebra and its Applications
publication date: 2003-05-25T00:00:00Z
review text: The author gives a very short proof of the Auslander defect formula, appearing for the first time in \textit{M. Auslander}'s Philadelphia notes [``Functors and morphisms determined by objects'', in: Representation theory of algebras, Proc. Conf. Philadelphia 1976, Lect. Notes Pure Appl. Math. 37, 1--244 (1978; Zbl 0383.16015)]. The formula relates the covariant and contravariant defect of a short exact sequence in a module category: if \(\delta:0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0\) is an exact sequence, then the covariant defect \(\delta_*\) and contravariant defect \(\delta^*\) are defined by the exact sequences \(0\rightarrow \Hom_R(C,-)\rightarrow \Hom_R(B,-)\rightarrow \Hom(A,-)\rightarrow \delta_*\) and \(0\rightarrow \Hom_R(-,A)\rightarrow \Hom_R(-,B)\rightarrow \Hom(-,C)\rightarrow \delta^*\). They are connected by the formula \(D\delta^*(X)\simeq \delta_*(D\,\text{Tr}X)\), where the transpose of \(X\), \(\text{Tr}X\), is defined by the exactness of the sequence \(P_0^*\rightarrow P_1^*\rightarrow \text{Tr}X \rightarrow 0\), for a finitely presented module \(X\) with a projective presentation \(P_1\rightarrow P_0\rightarrow X \rightarrow 0\). An immediate consequence of this formula is the classical Auslander-Reiten formula relating the functors Ext and Hom.
reviewed by: Constantin Năstăsescu
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 16D10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 1915494
zbMATH Keywords: short exact sequence
zbMATH Keywords: covariant defect
zbMATH Keywords: Auslander-Reiten formula
MaRDI profile type: Publication
cites work: Q4143219
cites work: Representation theory of artin algebras iii almost split sequences
cites work: Q4318498
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description: scientific article; zbMATH DE number 1915494