Maximal Buchsbaum modules of finite projective dimension (Q687629)

The purpose of the paper is to study the class of maximal Buchsbaum modules of finite projective dimension. The author first establishes an equivalence between the opposite category of the stable category of modules of finite projective dimension and a certain full subcategory of the derived category of complexes, which is very basic in the theory. Using this equivalence, the author classifies maximal Buchsbaum modules of finite projective dimension over a Gorenstein local ring. Let \({\mathcal B}\) be the stable category of all such modules. Then \({\mathcal B}\) contains a subcategory \({\mathcal B}'\) whose objects bijectively correspond to the representations of the quiver with two vertices and \(r\) arrows where \(r\) is an invariant of the local ring. The indecomposability is described in terms of positive roots associated with the generalized Cartan matrix \({\;\;2\;-r \choose -r\;\;\;2}\), and it is determined when \({\mathcal B}'\) and \({\mathcal B}\) are of finite representation type. The author also studies the class of maximal quasi-Buchsbaum modules of finite projective dimension [cf. \textit{M. Amasaki}, J. Math. Kyoto Univ. 33, No. 1, 143-170 (1993)]. On the other hand the author gives a corrected form of a theorem due to \textit{P. Schenzel} [cf. Adv. Math. 44, 61-77 (1982; Zbl 0492.13011); see also \textit{K. Yamagishi}, Math. Proc. Camb. Philos. Soc. 110, No. 2, 261- 279 (1991; Zbl 0760.13010)] and a proof of the structure theorem of maximal Buchsbaum modules due to \textit{S. Goto} [Commutative Algebra and Combinatorics, US-Jap. Joint Semin., Kyoto 1985, Adv. Stud. Pure Math. 11, 39-64 (1987; Zbl 0649.13009)].