A Hilbert-kunz Criterion for Solid Closure in Dimension Two (characteristic Zero)

Let I denote a homogeneous R+-primary ideal in a twodimensional 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 (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. Mathematical Subject Classification (2000): 13A35; 13D40; 14H60 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 q : f ∈ I) for a prime power q = p. Then the Hilbert-Kunz function of I is given by e 7−→ λ(R/I [p e) , where λ denotes the length. The Hilbert-Kunz multiplicity of I is defined as the limit eHK(I) = lim e→∞ λ(R/I [p e)/p . 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 q ∈ I [q] for almost all q = p} . 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