×

Primary decomposition over rings graded by finitely generated abelian groups. (English) Zbl 1139.13008

Let \(A\) be a commutative noetherian ring which is graded by a finitely generated abelian group \(G\). A result of Bourbaki says that if \(G\) is a torsion free, then the associated primes of finitely generated \(G\)-graded \(A\)-modules \(M\) are \(G\)-graded as well, and a \(G\)-graded primary decomposition of \(G\)-graded submodules of \(M\) does exist. In this paper the authors introduce in a natural manner the concept of a similar \(G\)-graded primary decomposition in case \(G\) has torsion, and show the existence of such decompositions.

MSC:

13E05 Commutative Noetherian rings and modules
13C99 Theory of modules and ideals in commutative rings
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bourbaki, N., Commutative Algebra (1989), Springer-Verlag
[2] Bergman, G., Everybody knows what a Hopf algebra is, (Contemp. Math., vol. 43 (1985)), 25-48 · Zbl 0569.16005
[3] Cohen, M.; Montgomery, S., Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc., 282, 1, 237-258 (1984) · Zbl 0533.16001
[4] Cox, D. A., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4, 1, 17-50 (1995) · Zbl 0846.14032
[5] Eisenbud, D.; Mustaţǎ, M.; Stillman, M., Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput., 29, 583-600 (2000) · Zbl 1044.14028
[6] Kumar, S. D., A note on a graded ring analogue of Quillen’s theorem, Expo. Math., 22, 297-298 (2004) · Zbl 1088.13500
[7] Năstăsescu, C.; van Oystaeyen, F., Methods of Graded Rings, Lecture Notes in Math., vol. 1836 (2004), Springer-Verlag
[8] M. Perling, G. Trautmann, Equivariant primary decomposition and toric sheaves, in preparation; M. Perling, G. Trautmann, Equivariant primary decomposition and toric sheaves, in preparation · Zbl 1196.14044
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.