Bounds for test exponents

2004/12/20 by Holger Brenner, Brenner, Holger · 1 citation
Mathematics · #13A35 #14D20 #14F05 #14H52 #14H60 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13A35 #msc:14D20 #msc:14F05 #msc:14H52 #msc:14H60

paper · pdf · doi:10.48550/arxiv.math/0412404

Some improvements. To appear in Compositio Math

arxiv created 2005/07/05 · arxiv updated 2009/12/01

Abstract

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) belongs to I^([p^b]) if and only if f belongs to the Frobenius closure I^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.

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• On the uniform bound of Frobenius test exponents
• Notes on the Frobenius test exponents
• Tight closure test exponents for certain parameter ideals
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• Frobenius test exponents for parameter ideals in generalized Cohen-Macaulay local rings