A short proof for Auslander's defect formula.
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.
scientific article; zbMATH DE number 1915494
scientific article