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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An excellent $ F$-pure ring of prime characteristic has a big tight closure test element


Author: Rodney Y. Sharp
Journal: Trans. Amer. Math. Soc. 362 (2010), 5455-5481
MSC (2010): Primary 13A35, 16S36, 13D45, 13E05, 13E10, 13H10; Secondary 13J10
Published electronically: May 3, 2010
MathSciNet review: 2657687
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Abstract: In two recent papers, the author has developed a theory of graded annihilators of left modules over the Frobenius skew polynomial ring over a commutative Noetherian ring $ R$ of prime characteristic $ p$, and has shown that this theory is relevant to the theory of test elements in tight closure theory. One result of that work was that, if $ R$ is local and the $ R$-module structure on the injective envelope $ E$ of the simple $ R$-module can be extended to a structure as a torsion-free left module over the Frobenius skew polynomial ring, then $ R$ is $ F$-pure and has a tight closure test element. One of the central results of this paper is the converse, namely that, if $ R$ is $ F$-pure, then $ E$ has a structure as a torsion-free left module over the Frobenius skew polynomial ring; a corollary is that every $ F$-pure local ring of prime characteristic, even if it is not excellent, has a tight closure test element. These results are then used, along with embedding theorems for modules over the Frobenius skew polynomial ring, to show that every excellent (not necessarily local) $ F$-pure ring of characteristic $ p$ must have a so-called `big' test element.


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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: http://dx.doi.org/10.1090/S0002-9947-10-05166-4
PII: S 0002-9947(10)05166-4
Keywords: Commutative Noetherian ring, prime characteristic, Frobenius homomorphism, tight closure, test element, $F$-pure ring, Frobenius skew polynomial ring
Received by editor(s): December 18, 2008
Published electronically: May 3, 2010
Article copyright: © Copyright 2010 American Mathematical Society