The membership of the local cohomology modules H n a (M) of a module M in certain Serre subcategories of the category of modules is studied from below (i < n) and from above (i > n). Generalizations of depth and regular sequences are defined. The relation of these notions to local cohomology are found. It is shown that the membership of the local cohomology modules of a finite module in a Serre subcategory in the upper range just depends on the support of the module.
Let I be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules T with dim T ≤ n and we show it by FD ≤n where n ≥ −1 is an integer. We prove that for any FD ≤0 (or minimax) submodule N of H t I (M ) the R-modules Hom R (R/I, H t I (M )/N ) and Ext 1 R (R/I, H t I (M )/N ) are finitely generated, whenever the modules H 0 I (M ), H 1 I (M ), ..., H t−1 I (M ) are FD ≤1 ( or weakly Laskerian). As a consequence, it follows that the associated primes of H t I (M )/N are finite. This generalizes the main results of Bahmanpour and Naghipour [4] and [5], Brodmann and Lashgari [7], Khashyarmanesh and Salarian [21] and Hong Quy [18]. We also show that the category F D 1 (R, I) cof of I-cofinite FD ≤1 R-modules forms an Abelian subcategory of the category of all R-modules.
Let R be a noetherian ring, a an ideal of R, and M an R-module. We prove that for a finite moduleWe give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.
Mathematics Subject Classification (2000). Primary 13D45; Secondary 13D07.
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