2016/01/31 by Rankeya Datta · 2 citations
Mathematics · #Algebra over a field #Algebraic structures and combinatorial models #Business #Cohomology #Commutative Algebra and Its Applications #De Rham cohomology #Equivariant cohomology #Ext functor #Finitely-generated abelian group #Group cohomology #Krull dimension #Local cohomology #Local ring #Mathematics #Motivic cohomology #Noetherian #Pure mathematics #Ring (chemistry) #Rings, Modules, and Algebras #Sheaf #Sheaf cohomology #Valuation (finance) #math.AC #msc:13D45 #msc:13F30 #msc:14F43 #Étale cohomology #Čech cohomology
paper · pdf · doi:10.1016/j.jalgebra.2016.12.032
published in Journal of Algebra 479, 413-436 (Elsevier BV) · Comments are welcome; latest edit corrects numerous typos and makes the article consistent with the journal version
openalex publication_date 2017/01/16 · arxiv created 2017/05/01 · arxiv updated 2017/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We examine local cohomology in the setting of valuation rings. The novelty of this investigation stems from the fact that valuation rings are usually non-Noetherian, whereas local cohomology has been extensively developed mostly in a Noetherian setting. We prove various vanishing results on local cohomology for valuation rings of finite Krull dimension. These vanishing results stem from a uniform bound on the global dimension of such rings. Our investigation reveals differences in the sheaf theoretic definition of local cohomology, and the algebraic definition in terms of a limit of certain Ext functors.