On the finiteness of local homology modules

Abstract

Let R be a commutative Noetherian ring and a an ideal of R. Let M be a finitely generated R-module and N an Artinian R-module. The concept of filter coregular sequence is introduced to determine the infimum of the integers i such that the generalized local homology H a i (M, N ) is not finitely generated as an R a -module, where R a denotes the a-adic completion of R. In particular, it is shown that H a i (M, N ) is a finitely generated R a -module for all i ∈ N 0 if and only if (0 : N a + Ann R (M )) has finite length whenever R is a complete semi-local ring.