1998/08/26 by Rolf Kaellstroem, Kaellstroem, Rolf · 3 citations
Mathematics · #14A10 #32C38 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG) #math.AG #math.QA #math.SG #msc:14A10 #msc:32C38
paper · pdf · doi:10.48550/arxiv.math/9808108
81 pages, Latex
arxiv created 1998/08/26 · openalex publication_date 1998/08/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Lie algebroid on a variety X/k is an extension α: gX → TX of the tangent sheaf both as OX-module and Lie algebra over the base field, with the obvious compatibilities; and given a Lie algebroid one has its associated ring of differential operators D(gX). Letting g'X be some Lie sub-algebroid of gX, a smooth coherent gX-module M is one in which any (locally defined) coherent OX-submodule M0 ⊂ M generates a g'X-submodule D(g'X)M0 that remains coherent over OX. Depending on the choice of g'X one gets in this way interesting classes of gX-modules, where probably the most important one is that of gX-modules with regular singularities. The paper is concerned with this notion of smoothness, including results about prolongation of smoothness over codimension ≥ 2, preservation of smoothness for inverse and direct images of gX-modules, and curve tests. In particular we treat gX-modules with regular singularities, generalizing some of the main theorems for D-modules with new and hopefully transparent proofs.