On the support of local cohomology

We examine the question of whether the support of an arbitrary local cohomology module of a finitely generated module over a Noetherian ring with support in a given ideal must be closed in the Zariski topology. Several results giving an affirmative answer to this question are given; in particular, we show that the support is closed when the given ideal has cohomological dimension at most two or the ambient ring is local of dimension at most four.