Hilbert-Kunz multiplicity of two-dimensional local rings (Q2731032)

Let \((A,\mathfrak m,k)\) denote a local Noetherian ring of characteristic \(p>0\) with \(d=\dim A\). Let \(M\) be a finitely generated \(R\)-module. For an \(\mathfrak m\)-primary ideal \(I\) the Hilbert-Kunz multiplicity \(e_{HK}(I, M)\) of \(M\) with respect to \(I\) is defined as the limit of \(L_A(M/I^{[q]}M)/q^d\), where \(q = p^e\). Recently there are various papers on the Hilbert-Kunz multiplicity [see e.g. \textit{K. Watanabe} and \textit{K. Yoshida}, J. Algebra 230, 295-317 (2000; Zbl 0964.13008) and the references quoted there]. The first main result of the present paper is a generalization of a result shown by \textit{C. Huneke} [see the book ``Tight closure and its application'', Reg. Conf. Ser. Math. 88 (1996; Zbl 0930.13004)]. It provides the following inequalities NEWLINE\[NEWLINE e(I^n)/d! \leq e_{HK}(I^n,A) \leq \binom{n+d-1}d e(I^n)/n^d,NEWLINE\]NEWLINE where \(e(I)\) denotes the ordinary multiplicity. Several applications of these bounds are discussed.NEWLINENEWLINENEWLINEFurthermore the authors study a weaker form of the following two conjectures:NEWLINENEWLINENEWLINE(1) \(e_{HK}(I,A) \geq L_A(A/I^\star),\) where \(I^\star\) denotes the tight closure of \(I\),NEWLINENEWLINENEWLINE(2) if \(A\) is Cohen-Macaulay, then \(e_{HK}(I,A) \geq L_A(A/I)\).NEWLINENEWLINENEWLINENamely, for a two-dimensional Cohen-Macaulay ring \(A\) it is shown that NEWLINE\[NEWLINEe_{HK}(\mathfrak m^n, A) \geq L_A(A/\mathfrak m^n)NEWLINE\]NEWLINE for all \(n\geq 1.\) If the equality holds for some \(n\geq 1\), then \(A\) is a regular local ring. This is a consequence of the lower bound \(e_{HK}(\mathfrak m^n,A) \geq e(A)/2 n^2 + n/2.\)NEWLINENEWLINENEWLINEAs a third main result the authors prove that rings with `minimal' Hilbert-Kunz multiplicity with respect to this bound are `Veronese subrings' in dimension 2.NEWLINENEWLINENEWLINEIn the final section the authors conclude with examples related to rational double points, Hilbert polynomials, and pseudo-rational local rings.