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The cokernel of the operator \(\partial/\partial x_ n\) acting on a \({\mathcal D}_ n\)-module. II. (English) Zbl 0579.58040

The author showed in part I of this paper [C. R. Acad. Sci., Paris, Sér. I 296, 903-906 (1983; Zbl 0536.58036)] that the cokernel of the operator \(\partial /\partial x_ n\) of a holonomic \({\mathcal D}_ n\)- module M is a holonomic \({\mathcal D}_{n-1}\)-module if M is so-called \(x_ n\)-regular.
In this paper we generalize this result to arbitrary \(x_ n\)-regular \({\mathcal D}_ n\)-modules, which are not necessary holonomic. In fact we show that for the category of \(x_ n\)-regular \({\mathcal D}_ n\)-modules \(d(\bar M)\leq d(M)-1\), where \(\bar M:=M/\partial_ nM\) and \(\partial_ n:= \partial/\partial x_ n\).

MSC:

58J99 Partial differential equations on manifolds; differential operators
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32L99 Holomorphic fiber spaces
47F05 General theory of partial differential operators
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References:

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