on the finiteness of local homology modules

Let r be a commutative noetherian ring and a be an ideal of r. suppose m is a finitely generated r-module and n is an artinian r-module. we define the concept of filter coregular sequence to determine the infimum of integers i such that the generalized local homology ha i (m, n) is not finitely generated as an ȓa -module, where ȓa denotes the a-adic completion of r. in particular, if r is a complete semi-local ring, then ha i (m, n) is a finitely generated ȓa -module for all non-negative integers i if and only if (0 :n a + ann(m)) has finite length.