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ARITHMETIC STRUCTURES FOR DIFFERENTIAL OPERATORS ON FORMAL SCHEMES

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

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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References

Ardakov, K., 𝓓̂-modules on rigid analytic spaces, Proc. Int. Congress of Mathematicians 2014 Seoul III (2014), 1–9.Google Scholar
Ardakov, K., Bode, A. and Wadsley, S. J., 𝓓̂-modules on rigid analytic spaces III. Preprint 2019, arXiv:1904.13280.10.1515/crelle-2016-0016CrossRefGoogle Scholar
Ardakov, K. and Wadsley, S., 𝓓̂-modules on rigid analytic spaces II: Kashiwara’s equivalence, J. Algebraic Geom. 27 (2018), 647–701.10.1090/jag/709CrossRefGoogle Scholar
Ardakov, K. and Wadsley, S., 𝓓̂-modules on rigid analytic spaces I, J. Reine Angew. Math. 747 (2019), 221–276.10.1515/crelle-2016-0016CrossRefGoogle Scholar
Ardakov, K. and Wadsley, S., On irreducible representations of compact p-adic analytic groups, Ann. of Math. (2) 178(2) (2013), 453–557.10.4007/annals.2013.178.2.3CrossRefGoogle Scholar
Berthelot, P., “Cohomologie rigide et théorie des 𝓓-modules”, in p-adic Analysis (Trento, 1989), Lecture Notes in Mathematics 1454, Springer, Berlin, 1990, 80–124.CrossRefGoogle Scholar
Berthelot, P., D-modules arithmétiques I. Opérateurs différentiels de niveau fini, Ann. Sci. ENS 29 (1996), 185–272.Google Scholar
Bosch, S., Güntzer, U. and Remmert, R., Non-Archimedean Analysis, Springer, Berlin, 1984.10.1007/978-3-642-52229-1CrossRefGoogle Scholar
Bosch, S., Lectures on Formal and Rigid Geometry, Lecture Notes in Math. 2105, Springer, Berlin, 2014.10.1007/978-3-319-04417-0CrossRefGoogle Scholar
Chiarellotto, B. and Le Stum, B., Pentes en cohomologie rigide et F-isocristaux unipotents, Manuscripta Math. 100(4) (1999), 455–468.10.1007/s002290050212CrossRefGoogle Scholar
Emerton, M., Locally analytic vectors in representations of locally p-adic analytic groups, Mem. Amer. Math. Soc. 248(1175) (2017), iv+158.Google Scholar
Fujiwara, K. and Kato, F., Foundations of Rigid Geometry. I, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2018.10.4171/135CrossRefGoogle Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. (32) (1967), 361.Google Scholar
Hartshorne, R., On the De Rham cohomology of algebraic varieties, Publ. Math. Inst. Hautes Études Sci. (45) (1975), 5–99.Google Scholar
Hartshorne, R., Residues and Duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Math. 20, Springer, Berlin–New York, 1966.10.1007/BFb0080482CrossRefGoogle Scholar
Hotta, R., Takeuchi, K. and Tanisaki, T., D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics 236, Birkhäuser Boston Inc., Boston, MA, 2008, Translated from the 1995 Japanese edition by Takeuchi.10.1007/978-0-8176-4523-6CrossRefGoogle Scholar
Huyghe, C., 𝒟sp†-affinité de l’espace projectif, Compos. Math. 108(3) (1997), 277–318; With an appendix by P. Berthelot.10.1023/A:1000124232370CrossRefGoogle Scholar
Huyghe, C., Patel, D., Schmidt, T. and Strauch, M., 𝓓† -affinity of formal models of flag varieties, Math. Res. Lett. (to appear) available at:https://perso.univ-rennes1.fr/tobias.schmidt/model_revised_ff_final.pdf.Google Scholar
Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics 6, Oxford University Press, Oxford, 2002, Translated from the French by Reinie Erné, Oxford Science Publications.Google Scholar
Noot-Huyghe, C., Finitude de la dimension homologique d’algèbres d’opérateurs différentiels faiblement complètes et à coefficients surconvergents, J. Algebra 307(2) (2007), 499–540.10.1016/j.jalgebra.2004.09.036CrossRefGoogle Scholar
Noot-Huyghe, C., Un théorème de Beilinson–Bernstein pour les 𝒟-modules arithmétiques, Bull. Soc. Math. France 137(2) (2009), 159–183.10.24033/bsmf.2572CrossRefGoogle Scholar
Patel, D., Schmidt, T. and Strauch, M., Integral models of ℙ1 and analytic distribution algebras for GL(2), Münster J. Math. 7 (2014), 241–271.Google Scholar
Patel, D., Schmidt, T. and Strauch, M., Locally analytic representations of GL(2, L) via semistable models of ℙ1, J. Inst. Math. Jussieu 18(2) (2019), 125–187.10.1017/S1474748016000396CrossRefGoogle Scholar
Schneider, P. and Teitelbaum, J., Algebras of p-adic distributions and admissible representations, Invent. Math. 153(1) (2003), 145–196.CrossRefGoogle Scholar
Shiho, A., Notes on generalizations of local Ogus–Vologodsky correspondence, J. Math. Sci. Univ. Tokyo 22(3) (2015), 793–875.Google Scholar
The Stacks Project Authors, Stacks Project. http://stacks.math.columbia.edu, 2017.Google Scholar