On the Hilbert-Samuel coefficients of Frobenius powers of an ideal

Abstract

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an m \mathfrak {m} -primary ideal exists in a Noetherian local ring ( R , m ) (R,\mathfrak {m}) with prime characteristic p > 0. p>0. This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring R R of a simplicial complex and an ideal J J generated by pure powers of the variables, the generalized Hilbert-Kunz function ℓ ( R / ( J [ q ] ) k ) \ell (R/(J^{[q]})^k) is a polynomial for all q , k q,k and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of J J in terms of Hilbert-Samuel multiplicity of J . J. We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal.