1 1 N ov 2 00 2 INDECOMPOSABLE CANONICAL MODULES AND CONNECTEDNESS

Throughout this paper all rings are commutative, with identity, and Noetherian, unless otherwise specified. In particular, “local ring” always means Noetherian local ring, unless otherwise specified. Our objective is to prove a generalization of Faltings’ connectedness theorem [Fal1, Fal2], which asserts that in a complete local domain (R,m,K) of dimension n, if I ⊆ m is an ideal generated by at most n−2 elements, then the punctured spectrum of R/I is connected. Our result (see Theorems 3.3 and 3.6) draws the same conclusion without the hypothesis that R be a domain: we assume instead that R is complete, equidimensional (i.e., for every minimal prime p of R, dimR/p = dimR), and that one of the following conditions, which we shall prove are equivalent, holds: a) H m(R) (local cohomology with support in m) is indecomposable. b) The canonical module ω of R is indecomposable. c) The S2-ification of R is local. d) For every ideal I of height two or more, Spec R− V (I) is connected. e) Given any two distinct minimal primes p, q of R, there is a sequence of minimal primes p = p0, . . . , pi, . . . , pr = q such that for 0 ≤ i < r, the height (pi + pi+1) ≤ 1.