On Hilbert-Kunz function and representation ring

摘要

Let p be a prime, (O, m) a complete local Z/(p)-algebra, and I_n the ideal of O generated by all a~p~n with a ∈ m. By e_n(O) we denote the length of O/I_n. The function n → e_n(O) is called the Hilbert-Kunz function of O. In this article, we deduce some multiplication formulae for the representation ring (Cf. §2, §3). By these formulae, we give a new proof of the main theorem in [4]: If O = F[|X_1, ..., X_t|]/(X_1~(d_1) + … + X_t~(d_t)) where d_i's are positive integers, then the Hilbert-Kunz function is n → cp~((t - 1)n) + △(n) where c is rational For the term △, there exist integers ω, l such that △(n + ω) = l△(n) for n o. We also obtain an explicit algorithm to evaluate e, l, ω and the function △ for any given prime p and d_i ≥ 1. By using the representation ring, we can also obtain the Hilbert-Kunz functions of some binomial hypersurfaces. (cf. §6)