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Zbl 0886.14004
Berthelot, Pierre
Arithmetic ${\cal D}$-modules. I: Differential operators of finite level. (${\cal D}$-modules arithmétiques. I: Opérateurs différentiels de niveau fini.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 29, No. 2, 185-272 (1996). ISSN 0012-9593

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.\par 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 $p$.
[G.Faltings (Bonn)]

MSC 2000:: *14F10 Special sheaves
14F30 p-adic cohomology

Keywords: differential operators; crystalline cohomology; divided powers

Cited in: Zbl 1200.14036 Zbl 1111.14006 Zbl 1126.14016 Zbl 1056.14025 Zbl 0948.14017 Zbl 0956.14010

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