2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65614The 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.Commutative Algebra13D45 (13E05 13H99 13J10)text