Hilbert-Kunz multiplicity and an inequality between multiplicity and colength
Let \((A,m)\) be a local ring of characteristic a prime number \(p>0\). The Hilbert-Kunz multiplicity of an \(m\)-primary ideal \(I\) is defined by the following limit: \(e_{HK}(I): =\lim_{e\to +\infty} {l_A(A/I^{[p^e] })\over p^{de}}\), where \(I^{[p^e]}\) is the ideal generated by all the \(p^e\)-th powers of the elements of \(I\). The main point of this paper consists of the proof that if \(A\) is an unmixed local ring of characteristic \(p>0\) with \(e_{HK}(I)=1\) then it is regular. The authors conjecture the following: Let \((A,m)\) be an unmixed local ring of characteristic a prime number \(p>0\), then: 1. For every parameter ideal \(Q\) of \(A\), \(e(Q)\geq l_A(A/Q^*)\), where \(Q^*\) is the tight closure of \(Q\) and \(e(Q)\) is the usual multiplicity; 2. If \(e(Q)=l_A (A/Q^*)\) for some parameter ideal \(Q\) of \(A\) then \(A\) is Cohen-Macaulay and \(F\)-rational. The authors investigate this conjecture with respect to some examples.
scientific article; zbMATH DE number 1523176
scientific article