ON THE FINITENESS OF LOCAL HOMOLOGY MODULES

Let R be a commutative Noetherian ring and \mathfrak{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 \textrm{H}^{\mathfrak{a}}_i(M, N) is not finitely generated as an \widehat{R}^{\mathfrak{a}}-module, where \widehat{R}^{\mathfrak{a}} denotes the \mathfrak{a}-adic completion of R. In particular, if R is a complete semi-local ring, then \textrm{H}^{\mathfrak{a}}_i(M, N) is a finitely generated \widehat{R}^{\mathfrak{a}}-module for all non-negative integers i if and only if (0:_N\mathfrak{a}+\textrm{Ann}(M)) has finite length.

ON THE FINITENESS OF LOCAL HOMOLOGY MODULES Keywords:

local homology , Tor functor , finiteness , filter coregular sequence