# 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

Available as: dvi dvi.gz ps ps.gz