Derived Functors and Hilbert Polynomials

Abstract

Let $R$ be a commutative Noetherian ring, $I$ an ideal, $M$ and $N$ finitely generated $R$-modules. Assume $V(I)\cap Supp(M)\cap Supp(N)$ consists of finitely many maximal ideals and let ${\l}(\e^i(N/I^nN,M))$ denote the length of $\e^i(N/I^nN,M)$. It is shown that ${\l}(\e^i(N/I^nN,M))$ agrees with a polynomial in $n$ for $n>>0$, and an upper bound for its degree is given. On the other hand, a simple example shows that some special assumption such as the support condition above is necessary in order to conclude that polynomial growth holds.