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.