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

Sharp, Rodney

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

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

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