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

Sharp, Rodney

doi:10.1090/S0002-9947-07-04247-X

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

Author: 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
MathSciNet review: 2309183
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Abstract: This paper is concerned with the tight closure of an ideal $\mathfrak{a}$ in a commutative Noetherian local ring of prime characteristic . 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 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 ; this theory is then applied in the later sections of the paper to the top local cohomology module of and used to show that, if is Cohen-Macaulay, then it must have a weak parameter test element, even if it is not excellent.

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