On the cofiniteness properties of certain general local cohomology modules

Let R be a commutative Noetherian ring, Phi a system of ideals of R, and M a finitely generated R-module. Suppose that a is an element of Phi and t is a non-negative integer. It is shown that if Ext(R)(i)(R/a, H-Phi(j) (M)) is finitely generated for all i and all j < t, then Ext(R)(i)(R/a, H-Phi(t) (M)) is finitely generated for i = 0, 1. In particular, if R is a local ring of dimension at most 2, then Ext(R)(i)(R/a, H-Phi(j) (M)) is finitely generated for all i, j.