Indecomposability of top local cohomology modules and Falting's finiteness dimension of modules

Let (R, m) be a commutative Noetherian local ring, I a proper ideal of R and M a finitely generated R-module of dimension d. We investigate Falting's finiteness dimension f(I)(M) and the equidimensionalness of certain quotients of M. It is seen, under some conditions, that f(I)(M) = max{1, grade(I, M)}. To achieve this, we first study the indecomposability of the top local cohomology module H-m(d)(M). As a consequence, in case Mis an indecomposable Cohen-Macaulay R-module, it turns out that Hdm(M) and therefore, when additionally R is complete, the canonical (dualizing) module K(M) are indecomposable. In this paper, we also find new results about the dimension of H-I(i)(M), i = 0, 1, 2, ... , d. Although dimR (HIM)-M-i) < d- iholds for all integers i, we show that there exists 0 = i = dsuch that dim(R)H(I)(i)(M) =d-i.