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

By R. Y. Sharp. 2006

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This paper is concerned with the tight closure of an ideal [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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/803de55f1142f00c92e96a4f16581a118c31be78

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Citations in the corpus

A survey of test ideals (2011, background): “n important object of study in commutative algebra for several decades. In particular, several questions about their ﬁniteness have been asked, and also answered, see for example [Ene03a], [EH08] and [Sha07]. These questions are closely related to the ﬁniteness of F-pure centers or compatibly φ-spli”
Centers of F-purity (2008, methodology): “One can obtain an alternate proof using the techniques of [ 35 ].”
The Frobenius structure of local cohomology (2007, result): “We would like to note here that other results related to F-stable submodules of local cohomology may be found in [Enescu 2001; 2003; Katzman 2006; Sharp 2007].”
p −1-Linear Maps in Algebra and Geometry (2012, result): “the following corollary which was obtained in [KM09] and also independently obtained by the second author in [Sch09]. In the case that X= SpecRand Ris local, proofs of this fact were ﬁrst obtained in [Sha07a] and [EH08]. Corollary 8.1.8. Let X be Frobenius split, then OX has only ﬁnitely many ideals”
Parameter-test-ideals of Cohen–Macaulay rings (2007, methodology): “ i ∩ T= (Bi ∩T) [pe] and Ci−1R= Bi−1 ⊆ Bi implies that Ci−1 = Ci−1R∩ T ⊆ Bi ∩ T and the minimality of Ci implies that Ci ⊆ Bi ∩ Thence CiR⊆ (Bi ∩ T)R⊆ Bi. 6. ES-ideals and special ES-ideals Following [S], we call an ideal K⊆ S[T;f] a graded two-sided ideal if K= M∞ i=0 KiT i for ideals K0,K1,... of ”
Cartier modules: Finiteness results (2009, result): “ow ourtheory implies global structural results which generalize analogous results for modules with a left Frobenius action over a local ring obtained recently by Enescu and Hochster [EH08], and Sharp [Sha07]. Our global version states: Theorem (Proposition 4.9). Let Xbe an F-ﬁnite scheme, and (M,C) ”