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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 eHK(I) = eHK((I, f)), where eHK 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 HilbertKunz criterion for tight closure in positive characteristic. Introduction Let (R, m) denote a local Noetherian ring or an N-graded algebra of dimension d of positive characteristic p. Let I denote an m-primary ideal, and set I [q] = (f : f ∈ I) for a prime power q = p. Then the Hilbert-Kunz function of I is given by e −→ λ(R/I [pe]) , where λ denotes the length. The Hilbert-Kunz multiplicity of I is defined as the limit eHK(I) = lim e→∞ λ(R/I [pe])/ped . This limit exists as a positive real number, as shown by Monsky in [9]. It is an open question whether this number is always rational. The Hilbert-Kunz multiplicity is related to the theory of tight closure. Recall that the tight closure of an ideal I in a Noetherian ring of characteristic p is by definition the ideal I∗={f ∈ R : ∃c not in any minimal prime : cf ∈ I [q] for almost all q = pe} . For an analytically unramified and formally equidimensional local ring R the equation eHK(I) = eHK(J) holds if and only if I∗ = J∗ holds true for ideals I ⊆ J (see [6, Theorem 5.4]). Hence f ∈ I∗ if and only if eHK(I) = eHK((I, f)). This is the Hilbert-Kunz criterion for tight closure in positive characteristic. The aim of this paper is to give a characteristic zero version of this relationship between Hilbert-Kunz multiplicity and tight closure for R+-primary homogeneous ideals in a normal two-dimensional graded domain R. There are several Received March 18, 2004. 2000 Mathematics Subject Classification. 13A35; 13D40; 14H60.

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