Bivariate Hilbert Functions for the Torsion Functor

Abstract

Let $(R,P)$ be a commutative, local Noetherian ring, $I$, $J$ ideals, $M$ and $N$ finitely generated $R$-modules. Suppose $J+an{n}_{R}M+an{n}_{R}N$ is $P$-primary. The main result of this paper is Theorem 6, which gives necessary and sufficient conditions for the length of $\t_i(M/I^nM,N/J^mN)$, to agree with a polynomial, for $m$, $n\gg 0$. As a corollary, it is shown that the length of $\t_i(M/I^nM,N/I^nN))$ always agrees with a polynomial in $n$, for $n\gg 0$, provided $I+an{n}_{R}M+an{n}_{R}N$ is $P$-primary.