FINITENESS DIMENSIONS AND COFINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

Let R be a commutative Noetherian ring with non-zero identity, a an ideal of R, M a finite R-module, and n a non-negative integer. In this paper, for an arbitrary R-module X which is not necessarily finite, we prove the following results: (i) f(a)(n) (M, X) = inf{i is an element of N-0 : H-a(i)(M, X) is not an FD<n R-module} if Ext(R)(i)(M/aM, X) is an FD<n R-module for all i; (ii) f(a)(1) (M, X) = inf{i is an element of N-0 : H-a(i)(M, X) is not a minimax R-module} if Ext(R)(i)(M/aM, X) is finite for all i; (iii) f(a)(2) (M, X) = inf{i is an element of N-0 : H-a(i)(M, X) is not a weakly Laskerian R-module} if R is semi-local and Ext(R)(i)(M/aM, X) is finite for all i; (iv) H-a(i)(M, X) is a-cofinite for all i < f(a)(2)(M, X) and Ass(R)(H-a(f2a(M,X)) (M, X)) is finite if Ext(R)(i)(M/aM, X) is finite for all i <= f(a)(2) (M, X). Here, f(a)(n)(M, X) = inf{f(aRp) (M-p, X-p) : p is an element of Spec(R) and dim(R)(R/p) >= n} is the nth finiteness dimension of M and X with respect to a and f(a)(M, X) = inf{i is an element of N-0 : H-a(i)(M, X) is not a finite R-module} is the finiteness dimension of M and X with respect to a.