ON THE COHOMOLOGICAL DIMENSION OF FINITELY GENERATED MODULES

Let (R, m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module Hom(R)(R/I,H-I(t) (R)) is finitely generated, then t = sup {dim <(<(R)over cap>(sB))over cap>/D : B is an element of V (I (R) over cap), D is an element of mAss(<(<(R)over cap>sB)over cap>)<(<(R)over cap>(sB))over cap> and B<(<(R)over cap>(sB))over cap> = Rad(I<(<(R)over cap>(sB))over cap> + D)}. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension R subset of R[X] will be included.