2017/12/25 by Mohammad Reza Doustimehr, Doustimehr, Mohammad Reza
Mathematics · #13D45 #13E05 #14B15 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1712.09067
openalex publication_date 2017/12/25 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
Let R be a commutative Noetherian ring, M a finitely generated R-module\nand n be a non-negative integer. In this article, it is shown that there is a\nfinitely generated submodule Ni of H frak ai(M) such that \dim rm\nSupp H frak ai(M)/Ni<n for all i<t if and only if there is a finitely\ngenerated submodule N_i, frak p of H_ frak a R frak pi(M frak\np) such that \dim rm Supp H_ frak a R frak pi(M frak\np)/N_i, frak p<n for all i<t. This generalizes Faltings' Local-global\nPrinciple for the finiteness of local cohomology modules (Faltings' in Math.\nAnn. 255:45-56, 1981). Also, it is shown that whenever R is a homomorphic\nimage of a Gorenstein local ring, then the invariants \inf i\∈ mathbb\nN0\|\dim rm Supp( frak btH frak ai(M))\≥ n for all \nt\∈ mathbb N0 and \inf rm depth M frak p+ rm ht( frak\na+ frak p)/ frak p\| frak p\∈ rm Spec R\∖ V( frak b)\n and \dim R/( frak a+ frak p) geqslant n are equal, for every\nfinitely generated R-module M and for all ideals frak a, frak b of R\nwith frak b\⊆ frak a. As a consequence, we determine the least\ninteger i where the local cohomology module H frak ai(M) is not minimax\n(resp. weakly laskerian).\n