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

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 $I^\star$ if and only if $e_{HK}(I) = e_{HK}((I,f))$, where $e_{HK}$ denotes the Hilbert-Kunz multiplicity of an ideal, introduced here in characteristic zero in the graded dimension two case. This provides a version in characteristic zero of the well-known Hilbert-Kunz criterion for tight closure in positive characteristic.

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Published 1 January 2004