  The purpose of this paper is to prove a generalization of Faltings'
connectedness theorem which asserts that, for a complete local domain R of
dimension n, the punctured spectrum of R/I is connected if the ideal I is
generated by at most n-2 elements. We replace the condition that R be a domain
by the requirement that the canonical module of R be indecomposable. We also
study equivalent conditions for the canonical module to be indecomposable;
under mild conditions this is equivalent to the S_2-ification of the local ring
to be local.
