Abstract :

Let R be a Noetherian local ring of characteristic p > 0 and I = (a_1, ..., a_r) be an ideal primary to the maximal ideal. Kunz proved that for large values of n, the colength of the ideal generated by p^n-th power of a_i's is a polynomial of degree equal to the dimension of R. The normalized coefficient of this polynomial is called the Hilbert-Kunz multiplicity of I. Kunz proved that for an ideal I, e_{HK}(I) =1 if and only if I is the maximal ideal and R is Regular Local Ring. We will discuss an elementary proof given by Huneke and Yao.