Indecomposable canonical modules and connectedness
  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. 
Melvin Hochster and Craig Huneke