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Ideals in a perfect closure, linear growth of primary decompositions, and tight closure


Authors: Rodney Y. Sharp and Nicole Nossem
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3687-3720
MSC (2000): Primary 13A35, 13E05, 13A15; Secondary 13B02, 13H05, 13F40, 13J10, 16S34, 16S36
DOI: https://doi.org/10.1090/S0002-9947-04-03420-8
Published electronically: January 13, 2004
MathSciNet review: 2055750
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Abstract: This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal ${\mathfrak{a}}$ of $R$has linear growth of primary decompositions, then tight closure (of ${\mathfrak{a}}$) `commutes with localization at the powers of a single element'. It is shown in this paper that, provided $R$ has a weak test element, linear growth of primary decompositions for other sequences of ideals of $R$ that approximate, in a certain sense, the sequence of Frobenius powers of ${\mathfrak{a}}$ would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of ${\mathfrak{a}}$) commutes with localization at an arbitrary multiplicatively closed subset of $R$.

Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of ${\mathfrak{a}}$has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of $R$, strategies for showing that tight closure (of a specified ideal ${\mathfrak{a}}$ of $R$) commutes with localization at an arbitrary multiplicatively closed subset of $R$and for showing that the union of the associated primes of the tight closures of the Frobenius powers of ${\mathfrak{a}}$is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman's question in the various situations considered are believed to be new.


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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

Nicole Nossem
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email: N.Nossem@sheffield.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-04-03420-8
Keywords: Commutative Noetherian ring, prime characteristic, Frobenius homomorphism, perfect closure, tight closure, plus closure, (weak) test element, primary decomposition, linear growth of primary decompositions, associated prime ideals, skew polynomial ring, skew Laurent polynomial ring, regular ring, pure subring, excellent rings
Received by editor(s): January 9, 2003
Received by editor(s) in revised form: May 15, 2003
Published electronically: January 13, 2004
Additional Notes: The first author was partially supported by the Engineering and Physical Sciences Research Council of the United Kingdom (Overseas Travel Grant Number GR/S11459/01) and the Mathematical Sciences Research Institute, Berkeley.
The second author was supported by a fees-only studentship provided by the Engineering and Physical Sciences Research Council of the United Kingdom.
Article copyright: © Copyright 2004 American Mathematical Society

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