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Bounds for test exponents

Published online by Cambridge University Press:  13 March 2006

Holger Brenner
Holger Brenner
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UKh.brenner@sheffield.ac.uk
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Abstract

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Suppose that R is a two-dimensional normal standard-graded domain over a finite field. We prove that there exists a uniform Frobenius test exponent b for the class of homogeneous ideals in R generated by at most n elements. This means that for every ideal I in this class we have that $f^{p^b}\in I^{[p^b]}$ if and only if $f\in I^{\rm F}$. This gives in particular a finite test for the Frobenius closure. On the other hand we show that there is no uniform bound for Frobenius test exponent for all homogeneous ideals independent of the number of generators. Under similar assumptions we prove also the existence of a bound for tight closure test ideal exponents for ideals generated by at most n elements.

Keywords

Frobenius closuretight closuretest exponentsemistable bundles13A3514D2014F0514H5214H60
Type
Research Article
Information
Compositio Mathematica , Volume 142 , Issue 2 , March 2006 , pp. 451 - 463
Copyright
Foundation Compositio Mathematica 2006