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Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure

Author(s): Rodney Y. Sharp
Journal: Trans. Amer. Math. Soc. 359 (2007), 4237-4258.
MSC (2000): Primary 13A35, 16S36, 13D45, 13E05, 13E10; Secondary 13H10
Posted: April 11, 2007
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Abstract: This paper is concerned with the tight closure of an ideal $ \mathfrak{a}$ in a commutative Noetherian local ring $ R$ of prime characteristic $ p$. Several authors, including R. Fedder, K-i. Watanabe, K. E. Smith, N. Hara and F. Enescu, have used the natural Frobenius action on the top local cohomology module of such an $ R$ to good effect in the study of tight closure, and this paper uses that device. The main part of the paper develops a theory of what are here called `special annihilator submodules' of a left module over the Frobenius skew polynomial ring associated to $ R$; this theory is then applied in the later sections of the paper to the top local cohomology module of $ R$ and used to show that, if $ R$ is Cohen-Macaulay, then it must have a weak parameter test element, even if it is not excellent.

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Additional Information:

Rodney Y. Sharp
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email: R.Y.Sharp@sheffield.ac.uk

DOI: 10.1090/S0002-9947-07-04247-X
PII: S 0002-9947(07)04247-X
Keywords: Commutative Noetherian ring, prime characteristic, Frobenius homomorphism, tight closure, (weak) test element, (weak) parameter test element, skew polynomial ring, local cohomology, Cohen--Macaulay local ring.
Received by editor(s): July 8, 2005
Posted: April 11, 2007
Additional Notes: The author was partially supported by the Engineering and Physical Sciences Research Council of the United Kingdom (Overseas Travel Grant Number EP/C538803/1).
Copyright of article: Copyright 2007, American Mathematical Society

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