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The continuity theorem of the division in the ring of differential operators. (Le théorème de continuité de la division dans les anneaux d’opérateurs différentiels.) (French) Zbl 0910.32011
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 \({\mathcal D}^\infty_X\). As a consequence, admissible modules over \({\mathcal D}_X^\infty\) and coherent modules over \({\mathcal D}_X\) inherit a canonical locally convex structure and admit finite free resolutions with strict morphisms. This structure allows, as example, to give a topological characterisation of regularity and to prove that the existence of a regular Bernstein-Sato functional equation for a coherent \({\mathcal D}_X\)-module, \({\mathcal M}\), with respect to an arbitrary divisor \(Y \subset X\), implies the comparison theorem \({\mathcal D}_X^\infty \otimes_{{\mathcal D}_X} {\mathcal M} [*Y] \simeq j_* j^{-1} {\mathcal M}^\infty\).
Reviewer: Z.Mebkhout (Paris)

32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials