×

Cohen-Macaulay and Gorenstein finitely graded rings. (English) Zbl 0655.13025

The main purpose of this paper is to characterize Cohen-Macaulay and Gorenstein graded rings of finite support (i.e. with only finitely many nonzero homogeneous components). One of the consequences of these characterizations is the following: If R is a strongly graded commutative ring over a finite group G, and e is the unit element of G, then R is Cohen-Macaulay (resp. Gorenstein) if and only if \(R_ e\) is so. The results are applied to the case of semitrivial extensions.
Reviewer: S.Raianu

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
16W50 Graded rings and modules (associative rings and algebras)
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] M.F. Atiyah - I.G. Macdonald , Introduction to Commutative Algebra , Addison-Wesley , London , 1969 . MR 242802 | Zbl 0175.03601 · Zbl 0175.03601
[2] H. Bass , On the ubiquity of Gorenstein rings , Math. Zeitschr. , 82 ( 1963 ) pp. 8 - 28 . MR 153708 | Zbl 0112.26604 · Zbl 0112.26604 · doi:10.1007/BF01112819
[3] M. Boratynski - S. Greco , When does an ideal arise from the Ferrand construction ?, Rapporto interno del Dipartimento di Matematica del Politecnico di Torino N. 2 , ( 1985 ).
[4] D. Ferrand , Courbes gauches et fibrés de rang 2 , C.R. Acad. Sci. Paris , 281 ( 1975 ), pp. 345 - 347 . MR 379517 | Zbl 0315.14019 · Zbl 0315.14019
[5] R. Fossum , Commutative extensions by canonical modules are Gorenstein rings , PAMS , 40 ( 1973 ), pp. 395 - 400 . MR 318139 | Zbl 0271.13013 · Zbl 0271.13013 · doi:10.2307/2039380
[6] R.M. Fossum - P.A. Griffith - I. Reiten , Trivial Extensions of Abelian Categories , LNM 456 , Springer-Verlag , Berlin - Heidelberg - New York , 1975 . MR 389981 | Zbl 0303.18006 · Zbl 0303.18006
[7] R. Hartshorne , Local Cohomology, LNM 41 , Springer-Verlag , Berlin - Heidelberg - New York , 1967 . MR 224620
[8] J. Herzog - E. Kunz , Der Kanonische Modul eines Cohen-Macaulay Rings , LNM 238 , Springer-Verlag , Berlin - Heidelberg - New York , 1973 . MR 412177 | Zbl 0231.13009 · Zbl 0231.13009 · doi:10.1007/BFb0059377
[9] I. Kaplansky , Commutative rings , The University of Chicago Press , Chicago and London , 1974 . MR 345945 | Zbl 0296.13001 · Zbl 0296.13001
[10] E. Matlis , Injective modules over Noetherian rings , Pacific J. Math. , 8 ( 1958 ), pp. 511 - 528 . Article | MR 99360 | Zbl 0084.26601 · Zbl 0084.26601 · doi:10.2140/pjm.1958.8.511
[11] E. Matlis , The Higher Properties of R-Sequences , J. of Algebra , 50 ( 1978 ), pp. 77 - 112 . MR 479882 | Zbl 0384.13002 · Zbl 0384.13002 · doi:10.1016/0021-8693(78)90176-X
[12] H. Matsumura , Commutative Algebra , Benjamin , New York , 1970 . MR 266911 | Zbl 0211.06501 · Zbl 0211.06501
[13] C. Menini , Finitely graded rings, Morita duality and self-injectivity , Comm. in Algebra , 15 ( 1987 ), pp. 1357 - 1364 . MR 898292 | Zbl 0618.18004 · Zbl 0618.18004 · doi:10.1080/00927878708823475
[14] C Năstăsescu , Strongly graded rings of finite groups , Comm. in Algebra , 11 ( 1983 ), pp. 1033 - 1071 . MR 700723 | Zbl 0522.16002 · Zbl 0522.16002 · doi:10.1080/00927872.1983.10487600
[15] C Năstăsescu - F. Van Oystaeyen , Graded Ring Theory , North-Holland , Amsterdam - New York - Oxford , 1982 . MR 676974 | Zbl 0494.16001 · Zbl 0494.16001
[16] I. Reiten , The converse to a theorem of Sharp on Gorenstein modules , Proc. of A.M.S. , 32 ( 1970 ), pp. 417 - 420 . MR 296067 | Zbl 0235.13016 · Zbl 0235.13016 · doi:10.2307/2037829
[17] J.P. Serre , Algebre Locale. Multiplicités , LNM 11 , Springer-Verlag , Berlin - Heidelberg - New York , 1965 . MR 201468 | Zbl 0142.28603 · Zbl 0142.28603
[18] R.Y. Sharp , Local Cohomology Theory in Commutative Algebra , Quart. J. Math. Oxford ( 2 ), 21 ( 1970 ), pp. 425 - 434 . MR 276217 | Zbl 0204.06003 · Zbl 0204.06003 · doi:10.1093/qmath/21.4.425
[19] R.Y. Sharp , Gorenstein Modules , Math. Z. , 115 ( 1970 ), pp. 117 - 139 . Article | MR 263801 | Zbl 0186.07403 · Zbl 0186.07403 · doi:10.1007/BF01109819
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.