On connectedness and indecomposibility of local cohomology modules

Let I denote an ideal of a local Gorenstein ring . Then we show that the local cohomology module , c = height I, is indecomposable if and only if V(I _d ) is connected in codimension one. Here I _d denotes the intersection of the highest dimensional primary components of I. This is a partial extension of a result shown by Hochster and Huneke in the case I the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of is a local Noetherian ring if dim R/I  =  1.