Abstract. Let R be a commutative Noetherian ring, I an ideal, M and N finitely generated R-modules. Assume V (I) ∩ Supp(M ) ∩ Supp(N ) consists of finitely many maximal ideals and let λ(Ext i (N/I n N, M )) denote the length of Ext i (N/I n N, M ). It is shown that λ(Ext i (N/I n N, 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.

Abstract. Let k be a field of characteristic 0, R = k[x 1 , . . . , x d ] be a polynomial ring, and m its maximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. Let λ(M) denote the length of an Rmodule M. In this paper, we show thatalways exists. This limit has been shown to be e(I)/d! for m-primary ideals I in a local Cohen-Macaulay ring, where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extension modules may not have polynomial growth.

Abstract. Let R be a local, Noetherian ring and I ⊆ R an ideal. A question of Kodiyalam asks whether for fixed i > 0, the polynomial giving the ith Betti number of I n has degree equal to the analytic spread of I minus one. Under mild conditions on R, we show that the answer is positive in a number of cases, including when I is divisible by m or I is an integrally closed m-primary ideal.

Abstract. Let (R, m) be a local ring and M and N finite R-modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules Ext i R (M, N/m n N ). A number of corollaries are given and more general filtrations are also considered.