×

Almost split sequences for Cohen-Macaulay-modules. (English) Zbl 0611.13009

Let R be a commutative complete local Gorenstein ring and \(\Lambda\) an R- algebra which is a finitely generated (maximal) Cohen-Macaulay R-module. Suppose C is nonprojective indecomposable in the category CM\(\Lambda\) consisting of the finitely generated \(\Lambda\) modules which are (maximal) Cohen-Macaulay R-modules. Our main result is that \(C_ p\) is \(\Lambda_ p\)-projective for all nonmaximal prime ideals P in R if and only if there is an almost split sequence \(0\to A\to B\to C\to 0\) in CM\(\Lambda\).

MSC:

13C10 Projective and free modules and ideals in commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D25 Complexes (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Auslander, M.: Functors and morphisms determined by objects. Proc. Conf. on Representation Theory, pp. 1-244 (Philadelphia 1976). New York: Dekker 1978
[2] Auslander, M.: Isolated singularities and almost split sequences, Proc. ICRA IV. Lect. Notes Math. 1178, 194-241. Berlin, Heidelberg, New York: Springer 1986 · Zbl 0633.13007
[3] Auslander, M., Reiten, I.: Representation theory of artin algebras. III. Almost split sequences. Commun. Algebra3, 239-294 (1975) · Zbl 0331.16027
[4] Auslander, M., Reiten, I.: Almost split sequences in dimension two. Adv. Math. (to appear) · Zbl 0625.13013
[5] Cartan, H., Eilenberg, S.: Homological algebra. Princeton: Princeton University Press 1956 · Zbl 0075.24305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.