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Truncations of the ring of number-theoretic functions

Jan Snellman

We study the ring \(\Gamma\) of all functions \(\Nat^+ \to K\), endowed with the usual convolution product. \(\Gamma\), which we call \emph{the ring of number-theoretic functions}, is an inverse limit of the ``truncations'' \[\Gamma_n = \setsuchas{f \in \Gamma}{\forall m > n: \, f(m)=0}.\] 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 \emph{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 Poincar\'e-Betti series for \(\Gamma_n\).

Homology, Homotopy and Applications, Vol. 2, 2000, No. 2, pp 17-27
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