Truncations of the ring of number-theoretic functions

Abstract

We study the ring Γ of all functions N + → K, endowed with the usual convolution product. Γ, which we call the ring of number-theoretic functions, is an inverse limit of the "truncations"Each Γ n is a zero-dimensional, finitely generated K-algebra, which may be expressed as the quotient of a finitely generated polynomial ring with a stable (after reversing the order of the variables) monomial ideal. Using the description of the free minimal resolution of stable ideals given by Eliahou-Kervaire, and some additional arguments by Aramova-Herzog and Peeva, we give the Poincaré-Betti series for Γ n .