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Local cohomology modules of invariant rings

Published online by Cambridge University Press:  18 December 2015

TONY J. PUTHENPURAKAL
TONY J. PUTHENPURAKAL*
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India. e-mail: tputhen@gmail.com
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Abstract

Let K be a field and let R be a regular domain containing K. Let G be a finite subgroup of the group of automorphisms of R. We assume that |G| is invertible in K. Let RG be the ring of invariants of G. Let I be an ideal in RG. Fix i ⩾ 0. If RG is Gorenstein then:

  1. (i) injdimRGHiI(RG) ⩽ dim Supp HiI(RG);

  2. (ii) $H^j_{\mathfrak{m}}$(HiI(RG)) is injective, where $\mathfrak{m}$ is any maximal ideal of RG;

  3. (iii) μj(P, HiI(RG)) = μj(P′, HiIR(R)) where P′ is any prime in R lying above P.

We also prove that if P is a prime ideal in RG with RGP not Gorenstein then either the bass numbers μj(P, HiI(RG)) is zero for all j or there exists c such that μj(P, HiI(RG)) = 0 for j < c and μj(P, HiI(RG)) > 0 for all j ⩾ c.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 2 , March 2016 , pp. 299 - 314
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1]Atiyah, M. F. and Macdonald, I. G.Introduction to Commutative Algebra (Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969).Google Scholar
[2]Bruns, W. and Herzog, J.Cohen–Macaulay rings (Cambridge University Press, 1993).Google Scholar
[3]Hartshorne, R.Affine duality and cofiniteness. Invent. Math. 9 (1969/1970), 145–164.CrossRefGoogle Scholar
[4]Huneke, C. and Sharp, R.Bass numbers of local cohomology modules. AMS Transactions. 339 (1993), 765–779.CrossRefGoogle Scholar
[5]Katzman, M.An example of an infinite set of associated primes of a local cohomology module. J. Algebra. 252 (2002), 161–166.CrossRefGoogle Scholar
[6]Lyubeznik, G.Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra) Invent. Math. 113 (1993), 41–55.CrossRefGoogle Scholar
[7]Lyubeznik, G.F-modules: applications to local cohomology and D-modules in characteristic p>0. J. Reine Angew. Math. 491 (1997) 65–130.CrossRefGoogle Scholar
[8]Matsumura, H.Commutative Ring Theory (Cambridge University Press, 1989).Google Scholar
[9]Singh, A.p-torsion elements in local cohomology modules. Math. Res. Lett. 7 (2000), 165–176.CrossRefGoogle Scholar
[10]Singh, A. and Swanson, I.Associated primes of local cohomology modules and of Frobenius powers. Int. Math. Res. Not. 33 (2004), 1703–1733.CrossRefGoogle Scholar
[11]Traves, W.Differential operators on orbifolds. J. Symbolic Comput. 41 (2006), 1295–1308.CrossRefGoogle Scholar
[12]Watanabe, K.Certain invariant subrings are Gorenstein. I. Osaka J. Math. 11 (1974), 1–8.Google Scholar