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.

Abstract. Let (R, m) be a local ring, and let M and N be finite R-modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules ExtA number of corollaries are given, and more general filtrations are also considered. §1. Introduction Let (R, m, k) be a Noetherian local ring, let I ⊆ R be an ideal, and let M and N be finitely generated R-modules. It is well known that if the lengths λ(M/I n M ) of the modules M/I n M are finite for n large, these lengths are given by a rational polynomial of degree dim(M ). In [7] (see also [6]) it is shown that the lengths of the modules Tor , where various assumptions were made in order to control this degree. In this paper we do not need to make any assumptions on M , N , or R to obtain our formulas, and we need only make modest assumptions on them to obtain a formula that makes direct reference only to M and N . In fact, in Section 2 we begin by giving a general

Abstract. Let R be local Noetherian ring of depth at least two. We prove that there are indecomposable R-modules which are free on the punctured spectrum of constant, arbitrarily large, rank.

Let (R, m, k) be a one-dimensional analytically unramified local ring with minimal prime ideals P 1 , . . . , Ps. Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the monoid C(R) of isomorphism classes of maximal Cohen-Macaulay R-modules: the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid give a measure of how badly this property fails. The key to understanding the monoid C(R) is determining the ranks of indecomposable maximal Cohen-Macaulay modules. Our main technical result shows that if R/P 1 has infinite Cohen-Macaulay type and the residue field k is infinite, then there exist |k| pairwise non-isomorphic indecomposable maximal Cohen-Macaulay R-modules of rank (r 1 , . . . , rs) provided r 1 ≥ r i for all i ∈ {1, . . . , s}. This result allows us to describe the monoid C(R) when R/Q has infinite Cohen-Macaulay type for every minimal prime ideal Q of the m-adic completion R.

a b s t r a c tComplete hypersurfaces of dimension at least 2 and multiplicity at least 4 have wild Cohen-Macaulay type.