Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure
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- by Rodney Y. Sharp PDF
- Trans. Amer. Math. Soc. 359 (2007), 4237-4258 Request permission
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.References
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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
- Received by editor(s): July 8, 2005
- Published electronically: 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 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 4237-4258
- MSC (2000): Primary 13A35, 16S36, 13D45, 13E05, 13E10; Secondary 13H10
- DOI: https://doi.org/10.1090/S0002-9947-07-04247-X
- MathSciNet review: 2309183