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1. Arithmetic structures for differential operators on formal schemes

Author

Huyghe, Christine, Schmidt, Tobias, and Strauch, Matthias

Subjects
Mathematics - Algebraic Geometry, 14G22, 16S32
Abstract

Let ${\mathfrak o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and ${\mathfrak X}_0$ a smooth formal scheme over the formal spectrum of ${\mathfrak o}$. Given an admissible formal blow-up ${\mathfrak X}$ of ${\mathfrak X}_0$ we introduce sheaves of differential operators ${\mathscr D}^\dagger_{{\mathfrak X},k}$ on ${\mathfrak X}$, for every integer $k \ge k_{\mathfrak X}$, where $k_{\mathfrak X}$ depends on the blow-up morphism ${\mathfrak X}\rightarrow {\mathfrak X}_0$. This generalizes Berthelot's construction of sheaves of arit hmetic differential operators on ${\mathfrak X}_0$. The coherence of these sheaves and several other basic properties are proven. In the second part we study the projective limit sheaf ${\mathscr D}_{{\mathfrak X},\infty} = \varprojlim_k {\mathscr D}^\dagger_{{\mathfrak X},k}$ and so-called coadmissible modules for ${\mathscr D}_{{\mathfrak X},\infty}$. The inductive limit of the sheaves ${\mathscr D}_{{\mathfrak X},\infty}$, over all admissible blow-ups ${\mathfrak X}$ of ${\mathfrak X}_0$, gives rise to a sheaf ${\mathscr D}_{\langle {\mathfrak X}_0 \rangle}$ on the Zariski-Riemann space of ${\mathfrak X}_0$. Analogues of Theorems A and B are shown to hold in each of these settings, i.e., for ${\mathscr D}^\dagger_{{\mathfrak X},k}$, ${\mathscr D}_{{\mathfrak X},\infty}$, and ${\mathscr D}_{\langle {\mathfrak X}_0\rangle}$., Comment: Some error corrected and some examples added

Published
2017
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2. Arithmetic structures for differential operators on formal schemes

Author

Christine Huyghe, Matthias Strauch, Tobias Schmidt, Institut de Recherche Mathématique Avancée ( IRMA ), Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Department of Mathematics, Indiana University, Indiana University [Bloomington], Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Indiana University System-Indiana University System, Indiana University System, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
General Mathematics, 010102 general mathematics, Formal scheme, Spectrum (functional analysis), Field (mathematics), 14G22, 16S32, Differential operator, 01 natural sciences, Discrete valuation ring, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, Fraction (mathematics), 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Arithmetic, [MATH]Mathematics [math], Mathematics::Representation Theory, Algebraic Geometry (math.AG), ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

Let ${\mathfrak o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and ${\mathfrak X}_0$ a smooth formal scheme over the formal spectrum of ${\mathfrak o}$. Given an admissible formal blow-up ${\mathfrak X}$ of ${\mathfrak X}_0$ we introduce sheaves of differential operators ${\mathscr D}^\dagger_{{\mathfrak X},k}$ on ${\mathfrak X}$, for every integer $k \ge k_{\mathfrak X}$, where $k_{\mathfrak X}$ depends on the blow-up morphism ${\mathfrak X}\rightarrow {\mathfrak X}_0$. This generalizes Berthelot's construction of sheaves of arit hmetic differential operators on ${\mathfrak X}_0$. The coherence of these sheaves and several other basic properties are proven. In the second part we study the projective limit sheaf ${\mathscr D}_{{\mathfrak X},\infty} = \varprojlim_k {\mathscr D}^\dagger_{{\mathfrak X},k}$ and so-called coadmissible modules for ${\mathscr D}_{{\mathfrak X},\infty}$. The inductive limit of the sheaves ${\mathscr D}_{{\mathfrak X},\infty}$, over all admissible blow-ups ${\mathfrak X}$ of ${\mathfrak X}_0$, gives rise to a sheaf ${\mathscr D}_{\langle {\mathfrak X}_0 \rangle}$ on the Zariski-Riemann space of ${\mathfrak X}_0$. Analogues of Theorems A and B are shown to hold in each of these settings, i.e., for ${\mathscr D}^\dagger_{{\mathfrak X},k}$, ${\mathscr D}_{{\mathfrak X},\infty}$, and ${\mathscr D}_{\langle {\mathfrak X}_0\rangle}$., Comment: Some error corrected and some examples added

Published
2017

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