Almost split sequences in dimension two
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 indecomposable \({\mathcal F}\) is locally free if and only if there exists such a sequence terminating at \({\mathcal F}\). Furthermore, such a sequence starts with \({\mathcal F}\otimes_{{\mathcal O}_ X}\omega_ X\), where \(\omega_ X\) is the dualizing sheaf, and when the rank of \({\mathcal F}\) is not divisible by the characteristic of k, the sequence is induced by tensoring with \({\mathcal F}^ a\) fixed almost split sequence \(0\to \omega_ X\to {\mathcal E}_ X\to {\mathcal O}_ X\to 0.\)
scientific article; zbMATH DE number 4014866