A surprising fact about D-modules in characteristic p>0

2004/07/27 by Josep Álvarez Montaner, Josep Alvarez Montaner, Montaner, Josep Alvarez +2
Mathematics · #13B30 #13N10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras #math.AC #math.AG #msc:13B30 #msc:13N10

paper · pdf · doi:10.48550/arxiv.math/0407464

8 pages

arxiv created 2004/07/27 · openalex publication_date 2004/07/27 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

Let R=k[x₁,...,x_d] be the polynomial ring in d independent variables, where k is a field of characteristic p>0. Let D be the ring of k-linear differential operators of R and let f be a polynomial in R. In this work we prove that the localization R[1/f] obtained from R by inverting f is generated as a D-module by 1/f. This is an amazing fact considering that the corresponding characteristic zero statement is very false.

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