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Action of finite groups on Rees algebras and Gorensteinness in invariant subrings

Published online by Cambridge University Press:  20 January 2009

Shin-Ichiro Iai
Affiliation:
Department of Mathematics School of Science and Technology, Meiji University, 214–71, Japan E-mail address: s-iai@math.meiji.ac.jp
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Let G be a finite group of order N and assume that G acts on a Cohen-Macaulay local ring A as automorphisms of rings. Let N be a unit in A. For a given G-stable ideal I in A we denote by R(I) = ⊕n≥0In and = G(I) = ⊕n≥0In/In+1 the Rees algebra and the associated graded ring of I, respectively. Then G naturally acts on R(I) and G(I) too. In this paper the conditions under which the invariant subrings R(I)G of R(I) are Cohen-Macaulay and/or Gorenstein rings are described in connection with the corresponding ring-theoretic properties of G(I)G and the a-invariants a(G(I)G of G(I)G. Consequences and some applications are discussed.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Aberbach, I. M., Huneke, C. and Smith, K. E., A tight closure approach to arithmetic Macaulayfication, preprint.Google Scholar
2.Atiyah, M. F. and Macdonald, I. G., Introduction to commutative algebra (Addison-Wesley, 1969).Google Scholar
3.Goto, S., On Gorenstein rings (in Japanese), Sûgatiu 31 (1979), 349364.Google Scholar
4.Goto, S. and Nishida, K., The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations (Mem. Amer. Math. Soc., vol. 526, Amer. Math. Soc., Providence, Rhode Island, 1994).Google Scholar
5.Goto, S. and Shimoda, Y., On the Rees algebras of Cohen-Macaulay local rings, in Commutative Algebra, Analytic Methods (Draper, R. N., ed., Lecture Notes in Pure and Applied Mathematics, vol. 68, Marcel Dekker, Inc., New York and Basel, 1982), 201231.Google Scholar
6.Goto, S. and Watanabe, K., On graded rings I, J. Math. Soc. Japan 30 (1978), 179213.Google Scholar
7.Herrman, M., Ikeda, S. and Orbanz, U., Equimultiplicity and blowing up. An algebraic study (Springer-Verlag, 1988).Google Scholar
8.Hochster, M. and Eagon, J. A., Cohen-Macaulay rings, invariant theory, and the generic perfection of determinational loci, Amer. J. Math. 93 (1971), 10201068.CrossRefGoogle Scholar
9.Ikeda, S., On the Gorensteinness of Rees algebras over local rings, Nagoya Math. J. 102 (1986), 135154.CrossRefGoogle Scholar
10.Watanabe, K., Certain invariant subrings are Gorenstein, I, Osaka J. Math. 11 (1974), 18.Google Scholar
11.Watanabe, K., Certain invariant subrings are Gorenstein, II, Osaka J. Math. 11 (1974), 379388.Google Scholar