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Arithmetic \({\mathcal D}\)-modules. I: Differential operators of finite level. (\({\mathcal D}\)-modules arithmétiques. I: Opérateurs différentiels de niveau fini.) (French) Zbl 0886.14004
The article deals with foundations of a theory of differential operators on a smooth scheme in mixed characteristics. A general problem in crystalline cohomology is the necessity to use factorials in denominators: Either by using DP-immersions and (dually) differential operators which are polynomials in the coordinate derivations, or by using nilpotent immersions and divided powers of coordinate derivations. In any case one of the relevant algebras becomes non noetherian.
In this work one uses a weakened version of divided powers (only a power \(x^q\), \(q=p^n\), has divided powers), then \(p\)-adically completes, inverts \(p\), and passes to the limit \(n\to\infty\). This gives a theory with coherent rings. To show this requires technical work, the key being that various intermediate rings are still noetherian, and various Tor-groups annihilated by a finite power of \(p\). This persists after \(p\)-adic completion, and transition maps become flat if one finally inverts \(p\).
Reviewer: G.Faltings (Bonn)

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F30 \(p\)-adic cohomology, crystalline cohomology
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