  It is a well-known result that, in projective space over a field, every
set-theoretical complete intersection of positive dimension in connected in
codimension one (Hartshorne [H1,3.4.6] or [H2, Theorem 1.3]). Another important
connectedness result is that a local ring with disconnected punctured sprectrum
has depth at most $1$ ([H1, Proposition 2.1]). The two results are related,
Hartshorne calls the latter "the keystone to the proof" of the former (loc.
cit).
  In this short note we show that 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 ([HeSc1], [HeSc2]). 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 cohomology module is the ring
itself ([HeSt, Theorem 2.2 (iii)]) and hence indecomposable as a module.
