Almost split sequences in dimension two
type: Item
instance of: scholarly article
title: Almost split sequences in dimension two
zbMATH Open document ID: 0625.13013
DOI: 10.1016/0001-8708(87)90031-4
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The results of this paper are closely related to those of two previous articles by \textit{M. Auslander} [Trans. Am. Math. Soc. 293, 511-531 (1986; Zbl 0594.20030) and ``Isolated singularities and almost split sequences'', in Representation theory I, Finite dimensional algebras, Proc. 4th Int. Conf., Ottawa/Can. 1984, Lect. Notes Math. 1178, 194-241 (1986)]. In the first part of the paper, the existence of almost split sequences in the category \underbar{CM}S of maximal Cohen-Macaulay modules over a complete two-dimensional Cohen-Macaulay local ring S is discussed. If C is an indecomposable module in \(\underline {CM}S\), and C is not free, it is proved that an almost split sequence \(0\to A\to B\to C\to 0\) exists if and only if C is free at height one primes. Furthermore, an exact sequence \(0\to \omega \to B\to S\to k\to 0\) is constructed, with \(\omega\) the dualizing module, \(B\in \underline{CM}S\), and k the residue field of S, which for any \(S_ 2\)-module M \((M\in \underline{CM}S)\), which is indecomposable, not free, and free at height one primes, produces an exact sequence \((*)\quad 0\to Hom_ S(M^*,\omega)\to Hom_ S(M^*,B)\to M\to 0,\) where \(M^*=Hom_ S(M,\omega).\) In this setup a trace function \(t: \Gamma = End_ S(M)\to S\) is defined, and in case it satisfies \(t(rad(\Gamma))\subset rad(S)\) (which is known to hold ife.g. S is a domain) (*) is an almost split sequence in \(\underline {CM}S\) if and only if \(t(\Gamma)\not\subset rad(S)\). These results were known earlier in the case of normal domains (loc. cit.). The second part of the paper establishes similar statements for two- dimensional positively graded rings \(S=(S_ i| i\in {\mathbb{Z}}),\) generated by \(S_ 1\) over the infinite field \(k=S_ 0\), and maximal Cohen-Macaulay graded S-modules. Sheafification yields results on the existence and structure of almost split sequences in the category of Cohen-Macaulay coherent sheaves over a connected projective Cohen- Macaulay curve X. An indecom…
reviewed by: Luchezar L. Avramov
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 16Gxx
zbMATH DE Number: 4014866
zbMATH Keywords: finite representation type
zbMATH Keywords: locally free sheaves
zbMATH Keywords: existence of almost split sequences
zbMATH Keywords: maximal Cohen-Macaulay modules
zbMATH Keywords: dualizing module
zbMATH Keywords: Cohen- Macaulay graded
zbMATH Keywords: dualizing sheaf
author: Maurice Auslander
author: Idun Reiten
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90031-4
OpenAlex ID: W1995830169
Wikidata QID: Q105554293
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description: scientific article; zbMATH DE number 4014866