On the vanishing and finiteness properties of generalized local cohomology modules

2009/09/07 by Moharram Aghapournahr, Aghapournahr, Moharram
Mathematics · #13D07 #13D45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #math.AC #math.AG #msc:13D07 #msc:13D45

paper · pdf · doi:10.48550/arxiv.0909.1131

5 pages

arxiv created 2009/09/07 · openalex publication_date 2009/09/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be a commutative noetherian ring, \fa an ideal of R and M,N finite R--modules. We prove that the following statements are equivalent. \beginenumerate \item[(i)] \lcⁱ_\fa(M,N) is finite for all i< n. \item[(ii)] \Coass_R(\lcⁱ_\fa(M,N)) ⊂ \V(\fa) for all i< n. \item[(iii)] \lcⁱ_\fa(M,N) is coatomic for all i< n. \endenumerate If \pd M is finite and r be a non-negative integer such that r>\pd M and \lcⁱ_\fa(M,N) is finite (resp. minimax) for all i≥ r, then \lcⁱ_\fa(M,N) is zero (resp. artinian) for all i≥ r.

Related

• On the vanishing, artinianness and finiteness of local cohomology modules
• On the Vanishing and the Finiteness of Supports of Generalized Local Cohomology Modules
• Some results on generalized local cohomology modules
• Cofiniteness of generalized local cohomology modules
• Some properties of generalized local cohomology modules