On the finiteness of local homology modules

2021/11/30 by Ali Fathi, Fathi, Ali, Alireza Hajikarimi +1
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2111.15337

Abstract

Let R be a commutative Noetherian ring and \mathfraka be an ideal of R. Suppose M is a finitely generated R-module and N is an Artinian R-module. We define the concept of filter coregular sequence to determine the infimum of integers i such that the generalized local homology \textrmH^\mathfraka_i(M, N) is not finitely generated as an \widehatR^\mathfraka-module, where \widehatR^\mathfraka denotes the \mathfraka-adic completion of R. In particular, if R is a complete semi-local ring, then \textrmH^\mathfraka_i(M, N) is a finitely generated \widehatR^\mathfraka-module for all non-negative integers i if and only if (0:_N\mathfraka+\textrmAnn(M)) has finite length.

Related

• Some properties of generalized local cohomology modules
• The finiteness dimension of local cohomology modules and its dual notion
• The local-global principle for the artinianness dimensions
• Local homology, finiteness of Tor modules and cofiniteness
• Dual of Faltings' Theorems on Finiteness of Local Cohomology