An excellent pure ring of prime characteristic has a big tight closure test element
Author:
Rodney Y. Sharp
Journal:
Trans. Amer. Math. Soc. 362 (2010), 54555481
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 of prime characteristic , 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 is local and the module structure on the injective envelope of the simple module can be extended to a structure as a torsionfree left module over the Frobenius skew polynomial ring, then is pure and has a tight closure test element. One of the central results of this paper is the converse, namely that, if is pure, then has a structure as a torsionfree left module over the Frobenius skew polynomial ring; a corollary is that every 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) pure ring of characteristic must have a socalled `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/S0002994710051664
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
