Truncations of the ring of number-theoretic functions

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 inverse limit of the “truncations” \[ \Gamma_n = \{ f \in \Gamma \mid \forall m > n: \, f(m)=0 \}. \] 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 $\Gamma_n$.

Keywords

ring of number theoretic functions, Poincaré-Betti series, stable monomial ideals

2010 Mathematics Subject Classification

13Dxx

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Published 1 January 2000