Hilbert-Kunz multiplicity and an inequality between multiplicity and colength

In this paper, we study local rings of small Hilbert-Kunz multiplicity. In particular, we prove that an unmixed local ring of Hilbert-Kunz multiplicity one is regular and classify two-dimensional Cohen-Macaulay local rings whose Hilbert-Kunz multiplicity is 2 or less. Also, we investigate the inequality between the multiplicity and the colength of the tight closure of parameter ideals inverse to the usual inequality between multiplicity and colength. (C) 2000 Academic Press.