Generalization of a connectedness result to cohomologically complete intersections

Abstract:

It is a well-known result from Hartshorne that, in projective space over a field, every set-theoretical complete intersection of positive dimension is connected in codimension one. Another important connectedness result (also from Hartshorne) is that a local ring with disconnected punctured spectrum has depth at most $1$. The two results are related; Hartshorne calls the latter “the keystone to the proof” of the former.

In this short note we show how the latter result generalizes smoothly from set-theoretical to cohomologically complete intersections, i.e., to ideals for which there is in terms of local cohomology no obstruction to be a complete intersection.

The proof is based on the fact that for cohomologically complete intersections over a complete local ring, the endomorphism ring of the (only) local cohomology module is the ring itself and hence is indecomposable as a module.