Filter Regular Sequences and Generalized Local Cohomology Modules

Abstract

Let a, b be ideals of a commutative Noetherian ring R and let M, N be finite R-modules. The concept of an a-filter grade of b on M is introduced and several characterizations and properties of this notion are given. Then, using the above characterizations, we obtain some results on generalized local cohomology modules H i a (M, N ). In particular, first we determine the least integer i for which H i a (M, N ) is not Artinian. Then we prove that H i a (M, N ) is Artinian for all i ∈ N 0 if and only if dim R/(a + Ann M + Ann N ) = 0. Also, we establish the Nagel-Schenzel formula for generalized local cohomology modules. Finally, in a certain case, the set of attached primes of H i a (M, N ) is determined and a comparison between this set and the set of attached primes of H i a (N ) is given.

Local-Global Principle for the Finiteness and Artinianness of Generalised Local Cohomology Modules

“…https://doi.org/10.1017/S0004972712000640 [8] Local-global principle on generalised local cohomology 487 …”

“…For each integer i, the ith generalised local cohomology functor H R (M/a n M, N) [2] Local-global principle on generalised local cohomology 481 has been studied in [12,15,17,21] and has led to some interesting results. We denote the common length of all maximal a-filter regular M-sequences contained in b by f-grad (a, b, M) and call it the a-filter grade of b on M. We briefly recall, in Section 2, the concept of a filter regular sequence and basic properties of f-grad (a, b, M), but refer the reader to [8,19] for more details. It is clear that an R-filter regular Msequence is just a weak M-sequence [2] and f-grad (R, b, M) = grad (b, M).…”