Truncations of the ring of number-theoretic functions

Abstract

We study the ring $\Gamma$ of all functions $\mathbb{N}^+ \to K$, endowed with the usual convolution product. $\Gamma$, which we call the ring of number-theoretic functions, is an adverse limit of the "truncations" $$\Gamma_n = \{ f \in \Gamma | \forall m > n : f(m) = 0 \}.$$ Each $\Gamma_n$ 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 $\Gamma_n$.