SOME PROPERTIES OF GENERALIZED LOCAL COHOMOLOGY MODULES

Amir Mafi

Abstract

          Let R be a commutative Noetherian ring, a an ideal of R, M and N be two finitely generated R-modules. Let t be a positive integer. We prove that if R is local with maximal ideal m and M_RN is of finite length then Hm^t(M,N) is of finite length for all t ≥ 0 and

lR(Hmt(M,N)) ≤

t ∑ i = 0

lR(ExtRi(M, Hmt-i(N)))

This yields l_R(Hm^t(M,N)) = l_R(Ext_R^t(M,N)).
           Additionally, we show that Ext_R^t(R/a,N) is Artinian for all  i ≤  t if and only if Haⁱ(M,N) is Artinian for all i ≤  t. Moreover, we show that whenever dim(R/a) = 0 then H_a^t(M,N) is Artinian for all t ≥ 0.