On the Connectedness of the Spectrum of Forcing Algebras

We study the connectedness property of the spectrum of forcing algebras over a noetherian ring. In particular we present for an integral base ring a geometric criterion for connectedness in terms of horizontal and vertical components of the forcing algebra. This criterion allows further simplifications when the base ring is local, or one--dimensional, or factorial. Besides, we discuss whether the connectedness of forcing algebras is a local property. Finally, we present a characterization of the integral closure of an ideal by means of the universal connectedness of the corresponding forcing morphism.