Local cohomology with support in a parameter ideal

Motivated in part by an attempt to understand better the notion of a parameter-like sequence introduced in Hochster (2003) [7, (2.2)], , we study results concerning the heights of the annihilators and the finiteness of the dimension of the socle in certain local cohomology modules with support in a parameter ideal. We obtain positive results under certain hypotheses of low dimension or codimension, but we also find examples that show, in the general case, that support in a parameter ideal does not restrict the behavior of local cohomology much more than support in an arbitrary ideal. The results obtained here strongly suggest that it would be worthwhile to seek a modification of the notion of a parameter-like sequence introduced in [7].