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Continuous division of linear differential operators and faithful flatness of $${\mathcal D}^\infty_X$$ over $${\mathcal D}_X$$. (English) Zbl 1061.32008
Elements of the theory of geometric differential systems. Papers from the C.I.M.P.A summer school, Séville, Spain, September 1996. Paris: Société Mathématique de France (ISBN 2-85629-151-1/pbk). Séminaires et Congrès 8, 129–148 (2004).
The authors give an elementary self-contained proof of the faithful flatness of the sheaf of differential operators of infinite order over the sheaf of differential operators of finite order. As a first step, they consider the ring of differential operators of infinite order as the completion of the corresponding ring of finite order ones for a natural topology. As a second step, they mimic Serre’s proof of the faithful flatness of the completion of a noetherian local ring over the ring itself. Here, the essential technical tool is the continuity of the Weierstrass-Grauert-Hironaka division of differential operator. They reproduce here the proof of H. Hauser and L. Narváez-Macarro given in Ann. Inst. Fourier 51, No. 3, 769–778 (2001; Zbl 0977.32009), which simplifies that of Z. Mebkhout and L. Narváez-Macarro given in J. Reine Angew. Math. 503, 193–236 (1998; Zbl 0910.32011).
For the entire collection see [Zbl 1050.32001].

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
##### Keywords:
infinite order differential operator; division theorem