We study the number of generators of ideals in regular rings and ask the question whether µ(I) < µ(I 2 ) if I is not a principal ideal, where µ(J) denotes the number of generators of an ideal J. We provide lower bounds for the number of generators for the powers of an ideal and also show that the CM-type of I 2 is ≥ 3 if I is a monomial ideal of height n in K[x 1 , . . . , x n ] and n ≥ 3.2010 Mathematics Subject Classification. Primary 13C99; Secondary 13H05, 13H10.
In this paper we study local duality and the vanishing and non-vanishing of generalized local cohomology. In particular for a local ring R and a finitely generated R-module N, the category N ⊥ of finitely generated R-modules M with H i m (M, N ) = 0 for all i = depth N is studied.
We study, in certain cases, the notions of finiteness and stability of the set of associated primes and vanishing of the homogeneous pieces of graded generalized local cohomology modules.
Let R = i 0 R i be a homogeneous Noetherian ring with local base ring (R 0 , m 0 ) and let M, N be two finitely generated graded R-modules. Let H i R+ (M, N ) denote the i-th graded generalized local cohomology of N relative to M with support in R + = i 1 R i . We study the vanishing, tameness and asymptotical stability of the homogeneous components of H i R+ (M, N ).
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