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ARITHMETIC STRUCTURES FOR DIFFERENTIAL OPERATORS ON FORMAL SCHEMES

Part of: Arithmetic problems. Diophantine geometry Rings and algebras arising under various constructions

Published online by Cambridge University Press:  19 December 2019

CHRISTINE HUYGHE ,
TOBIAS SCHMIDT Open the ORCID record for TOBIAS SCHMIDT [Opens in a new window]  and
MATTHIAS STRAUCH Open the ORCID record for MATTHIAS STRAUCH [Opens in a new window]
CHRISTINE HUYGHE
Affiliation:
IRMA, Université de Strasbourg, 7 rue René Descartes, 67084Strasbourg cedex, France email [email protected]
TOBIAS SCHMIDT
Affiliation:
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000Rennes, France email [email protected]
MATTHIAS STRAUCH
Affiliation:
Indiana University, Department of Mathematics, Rawles Hall, Bloomington, IN 47405, USA email [email protected]
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Abstract

Let $\mathfrak{o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and $\mathfrak{X}_{0}$ a smooth formal $\mathfrak{o}$-scheme. Let $\mathfrak{X}\rightarrow \mathfrak{X}_{0}$ be an admissible blow-up. In the first part, we introduce sheaves of differential operators $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ on $\mathfrak{X}$, for every sufficiently large positive integer $k$, generalizing Berthelot’s arithmetic differential operators on the smooth formal scheme $\mathfrak{X}_{0}$. The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf $\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves $\mathscr{D}_{\mathfrak{X},\infty }$, over all admissible blow-ups $\mathfrak{X}$, is a sheaf $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ on the Zariski–Riemann space of $\mathfrak{X}_{0}$, which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$, $\mathscr{D}_{\mathfrak{X},\infty }$, and $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$.

MSC classification

Primary: 14G22: Rigid analytic geometry 16S32: Rings of differential operators
Type
Article
Information
Nagoya Mathematical Journal , Volume 243 , September 2021 , pp. 157 - 204
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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