Local homology, finiteness of Tor modules and cofiniteness

2017/01/26 by Kamran Divaani-Aazar, Divaani-Aazar, Kamran, Hossein Faridian +3
Mathematics · #13D45 #13E05 #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.1701.07721

openalex publication_date 2017/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \frak a be an ideal of a commutative noetherian ring R with unity and M an R-module supported at \V(\fa). Let n be the supermum of the integers i for which H^\fa_i(M)≠ 0. We show that M is \fa-cofinite if and only if the R-module \Tor^R_i(R/\fa,M) is finitely generated for every 0≤ i≤ n. This provides a hands-on and computable finitely-many-steps criterion to examine \mathfraka-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.

Related

• Some results on local cohomology modules
• Cofiniteness of generalized local cohomology modules
• Associated primes and cofiniteness of local cohomology modules
• Cofiniteness of local cohomology modules and subcategories of modules
• Cofiniteness of generalized local cohomology modules for ideals of small dimension