Constructing big indecomposable modules

Abstract

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.

Hilbert-Samuel polynomials for the contravariant extension functor

“…Let (R, m, k) be a local ring and M a finitely generated R-module, free of constant rank on the punctured spectrum of R. Assume dim(R) ≥ 2, that the betti numbers of M are non-decreasing and that i < p. d.(M ). Then dim(Ω i+1 R (M )) = d. Our second purpose, especially in regards to section three, is to lay the groundwork for results concerning indecomposable modules in [2], where knowledge of the relative growth of the Hilbert polynomials of large syzygies of the residue field k is required. In particular, Theorem 1.1 above plays a crucial role in [2].…”

Hilbert-Samuel polynomials for the contravariant extension functor

“…Consequently, as an application of our degree formula, we obtain the following proposition, which yields some information about the dimension of the syzygies of finite-length modules. In Proposition 1.2, we use p. Our second purpose, especially regarding Section 3, is to lay the groundwork for results concerning indecomposable modules in [2], where knowledge of the relative growth of the Hilbert polynomials of large syzygies of the residue field k is required. In particular, Theorem 1.1 above plays a crucial role in [2].…”