2015/02/04 by Waqas Mahmood, Mahmood, Waqas
Mathematics · #13D45 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AC #msc:13D45
paper · pdf · doi:10.48550/arxiv.1502.01108
17 pages
openalex publication_date 2015/02/04 · arxiv created 2017/01/18 · arxiv updated 2017/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let I be an ideal of a commutative Noetherian ring R. Let M and N be any R-modules. We define the generalized completion homology modules LiΛI (N,M), for i∈ ℤ, as the homologies of the complex lim\longleftarrow(N/IsN⊗R F⋅R). Here F⋅R denote a flat resolution of M. In this article we will prove the vanishing and non-vanishing properties of LiΛI (N,M). We denote HiI(N,M) (resp. UIi(N,M)) by the generalized local cohomology modules (resp. the generalized local homology modules). As a technical tool we will construct several natural homomorphisms of LiΛI (N,M), HiI(N,M) and UIi(N,M). We will investigate when these natural homomorphisms are isomorphisms. Moreover if M is Artinian and N is finitely generated then it is proven that LiΛI (N,M) is isomorphic to UIi(N,M) for each i∈ ℤ. The similar result is obtained for HiI(N,M). Furthermore if both M and N are finitely generated with c=\grade(I,M). Then we are able to prove several necessary and sufficient conditions such that HiI(M)=0 for all i≠ c. Here HiI(M) denote the ordinary local cohomology module.