A Hilbert-Kunz criterion for solid closure in dimension two (characteristic zero)

2004/03/17 by Holger Brenner, Brenner, Holger
Mathematics · #13A35 #13D40 #14H60 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13A35 #msc:13D40 #msc:14H60

paper · pdf · doi:10.48550/arxiv.math/0403285

12 pages

arxiv created 2004/03/17 · arxiv updated 2009/12/01

Abstract

Let I denote a homogeneous R₊-primary ideal in a two-dimensional normal standard-graded domain over an algebraically closed field of characteristic zero. We show that a homogeneous element f belongs to the solid closure I^* if and only if e_HK(I) = e_HK((I,f)), where e_HK denotes the (characteristic zero) Hilbert-Kunz multiplicity of an ideal. This provides a version in characteristic zero of the well-known Hilbert-Kunz criterion for tight closure in positive characteristic.

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