2012/10/11 by Luis Núñez‐Betancourt, Luis Núñez-Betancourt, Núñez-Betancourt, Luis
Mathematics · #13D45 #13N10 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AC #msc:13D45 #msc:13N10
paper · pdf · doi:10.48550/arxiv.1210.3107
15 pages
arxiv created 2012/10/11 · openalex publication_date 2012/10/11 · arxiv updated 2012/10/12 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
In this article, we study the following question raised by Mel Hochster: let (R,m,K) be a local ring and S be a flat extension with regular closed fiber. Is \cV(mS)∩\AssS HiI(S) finite for every ideal I⊂ S and i∈ \NN? We prove that the answer is positive when S is either a polynomial or a power series ring over R and dim(R/I∩ R)≤ 1. In addition, we analyze when this question can be reduced to the case where S is a power series ring over R. An important tool for our proof is the use of Σ-finite D-modules, which are not necessarily finitely generated as D-modules, but whose associated primes are finite. We give examples of this class of D-modules and applications to local cohomology.