Le théorème de continuité de la division dans les anneaux d'opérateurs différentiels

Abstract

In this paper we prove the continuity of Weierstrass-Hironaka division of finite order linear differential operators over a complex analytic manifold X with respect to the induced topology by a canonical one of Fréchet nuclear on the sheaf script D∞X. As a consequence, admissible modules over script D∞X and coherent modules over script DX inherit a canonical locally convex structure and admit finite free resolutions with strict morphisms. This structure allows, as example, to give a topological caracterisation of regularity and to prove that the existence of a regular Bernstein-Sato functional equation for a coherent script DX-module, M, with respect to an arbitrary divisor Y ⊂ X, implies the comparison theorem script D∞X ⊗script DX M[*Y] ≃ j*j-1 M∞.