$\wideparen {\mathcal {D}}$-modules on rigid analytic spaces II: Kashiwaraâs equivalence
Authors:
Konstantin Ardakov and Simon Wadsley
Journal:
J. Algebraic Geom. 27 (2018), 647-701
DOI:
https://doi.org/10.1090/jag/709
Published electronically:
July 19, 2018
MathSciNet review:
3846550
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Abstract |
References |
Additional Information
Abstract: Let $X$ be a smooth rigid analytic space. We prove that the category of co-admissible $\wideparen {\mathcal {D}_X}$-modules supported on a closed smooth subvariety $Y$ of $X$ is naturally equivalent to the category of co-admissible $\wideparen {\mathcal {D}_Y}$-modules and use this result to construct a large family of pairwise non-isomorphic simple co-admissible $\wideparen {\mathcal {D}_X}$-modules.
References
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- K. Ardakov and S. J. Wadsley, $\wideparen {\mathcal {D}}$-modules on rigid analytic spaces I, J. Reine Angew. Math., 2017.
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- Oren Ben-Bassat and Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 1, 49â126 (English, with English and French summaries). MR 3626003, DOI https://doi.org/10.5802/afst.1526
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- Pierre Berthelot, Introduction Ă la thĂ©orie arithmĂ©tique des $\scr D$-modules, AstĂ©risque 279 (2002), 1â80 (French, with French summary). Cohomologies $p$-adiques et applications arithmĂ©tiques, II. MR 1922828
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- Ryuichi Ishimura, Homomorphismes du faisceau des germes de fonctions holomorphes dans lui-mĂȘme et opĂ©rateurs diffĂ©rentiels, Mem. Fac. Sci. Kyushu Univ. Ser. A 32 (1978), no. 2, 301â312 (French). MR 509326, DOI https://doi.org/10.2206/kyushumfs.32.301
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- Masaki Kashiwara and Takahiro Kawai, On holonomic systems of microdifferential equations. III. Systems with regular singularities, Publ. Res. Inst. Math. Sci. 17 (1981), no. 3, 813â979. MR 650216, DOI https://doi.org/10.2977/prims/1195184396
- Mark Kisin, Local constancy in $p$-adic families of Galois representations, Math. Z. 230 (1999), no. 3, 569â593. MR 1680032, DOI https://doi.org/10.1007/PL00004706
- Huishi Li and Freddy van Oystaeyen, Zariskian filtrations, $K$-Monographs in Mathematics, vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. MR 1420862
- Z. Mebkhout, Une Ă©quivalence de catĂ©gories, Compositio Math. 51 (1984), no. 1, 51â62 (French). MR 734784
- Zoghman Mebkhout, ThĂ©orĂšmes de bidualitĂ© locale pour les ${\cal D}_{X}$-modules holonomes, Ark. Mat. 20 (1982), no. 1, 111â124 (French). MR 660129, DOI https://doi.org/10.1007/BF02390502
- Fabienne Prosmans and Jean-Pierre Schneiders, A topological reconstruction theorem for $\scr D^\infty $-modules, Duke Math. J. 102 (2000), no. 1, 39â86. MR 1741777, DOI https://doi.org/10.1215/S0012-7094-00-10212-8
- George S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195â222. MR 154906, DOI https://doi.org/10.1090/S0002-9947-1963-0154906-3
- Mikio Sato, Takahiro Kawai, and Masaki Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of AndrĂ© Martineau), Springer, Berlin, 1973, pp. 265â529. Lecture Notes in Math., Vol. 287. MR 0420735
- Peter Schneider and Jeremy Teitelbaum, Algebras of $p$-adic distributions and admissible representations, Invent. Math. 153 (2003), no. 1, 145â196. MR 1990669, DOI https://doi.org/10.1007/s00222-002-0284-1
- The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2014.
References
- K. Ardakov and O. Ben-Bassat, Bounded linear endomorphisms of rigid analytic functions, Proc. London Math. Soc. (to appear), DOI 10.1112/plms.12142.
- K. Ardakov and S. J. Wadsley, $\wideparen {\mathcal {D}}$-modules on rigid analytic spaces I, J. Reine Angew. Math., 2017.
- A. A. BeÄlinson and V. V. Schechtman, Determinant bundles and Virasoro algebras, Comm. Math. Phys. 118 (1988), no. 4, 651â701. MR 962493
- Oren Ben-Bassat and Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 1, 49â126 (English, with English and French summaries). MR 3626003, DOI https://doi.org/10.5802/afst.1526
- Pierre Berthelot, ${\mathcal {D}}$-modules arithmĂ©tiques. I. OpĂ©rateurs diffĂ©rentiels de niveau fini, Ann. Sci. Ăcole Norm. Sup. (4) 29 (1996), no. 2, 185â272 (French, with English summary). MR 1373933
- Pierre Berthelot, Introduction Ă la thĂ©orie arithmĂ©tique des $\mathcal {D}$-modules, Cohomologies $p$-adiques et applications arithmĂ©tiques, II, AstĂ©risque 279 (2002), 1â80 (French, with French summary). MR 1922828
- Jan-Erik Björk, Analytic ${\mathcal {D}}$-modules and applications, Mathematics and its Applications, vol. 247, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1232191
- S. Bosch, U. GĂŒntzer, and R. Remmert, Non-Archimedean analysis, A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. MR 746961
- Siegfried Bosch, Werner LĂŒtkebohmert, and Michel Raynaud, Formal and rigid geometry. III. The relative maximum principle, Math. Ann. 302 (1995), no. 1, 1â29. MR 1329445, DOI https://doi.org/10.1007/BF01444485
- Brian Conrad, Several approaches to non-Archimedean geometry, $p$-adic geometry, Univ. Lecture Ser., vol. 45, Amer. Math. Soc., Providence, RI, 2008, pp. 9â63. MR 2482345, DOI https://doi.org/10.1090/ulect/045/02
- Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, translated from the 1995 Japanese edition by Takeuchi, Progress in Mathematics, vol. 236, BirkhÀuser Boston, Inc., Boston, MA, 2008. MR 2357361
- JoS (http://mathoverflow.net/users/69630/jos). When is a map from a logarithmic tangent bundle to a normal bundle surjective? MathOverflow, http://mathoverflow.net/q/231300 (version: 2016-02-17).
- Johannes Huebschmann, Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103â159. MR 1696093, DOI https://doi.org/10.1515/crll.1999.043
- C. Huyghe, D. Patel, T. Schmidt, and M. Strauch, $\mathscr {D}^{\dag }$-affinity of formal models of flag varieties, arXiv e-prints, January 2015.
- Ryuichi Ishimura, Homomorphismes du faisceau des germes de fonctions holomorphes dans lui-mĂȘme et opĂ©rateurs diffĂ©rentiels, Mem. Fac. Sci. Kyushu Univ. Ser. A 32 (1978), no. 2, 301â312 (French). MR 509326, DOI https://doi.org/10.2206/kyushumfs.32.301
- Masaki Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), no. 2, 319â365. MR 743382, DOI https://doi.org/10.2977/prims/1195181610
- Masaki Kashiwara and Takahiro Kawai, On holonomic systems of microdifferential equations. III. Systems with regular singularities, Publ. Res. Inst. Math. Sci. 17 (1981), no. 3, 813â979. MR 650216, DOI https://doi.org/10.2977/prims/1195184396
- Mark Kisin, Local constancy in $p$-adic families of Galois representations, Math. Z. 230 (1999), no. 3, 569â593. MR 1680032, DOI https://doi.org/10.1007/PL00004706
- Huishi Li and Freddy van Oystaeyen, Zariskian filtrations, $K$-Monographs in Mathematics, vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. MR 1420862
- Z. Mebkhout, Une Ă©quivalence de catĂ©gories, Compositio Math. 51 (1984), no. 1, 51â62 (French). MR 734784
- Zoghman Mebkhout, ThĂ©orĂšmes de bidualitĂ© locale pour les ${\mathcal {D}}_{X}$-modules holonomes, Ark. Mat. 20 (1982), no. 1, 111â124 (French). MR 660129, DOI https://doi.org/10.1007/BF02390502
- Fabienne Prosmans and Jean-Pierre Schneiders, A topological reconstruction theorem for $\mathcal {D}^\infty$-modules, Duke Math. J. 102 (2000), no. 1, 39â86. MR 1741777, DOI https://doi.org/10.1215/S0012-7094-00-10212-8
- George S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195â222. MR 0154906, DOI https://doi.org/10.2307/1993603
- Mikio Sato, Takahiro Kawai, and Masaki Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of AndrĂ© Martineau), Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973, pp. 265â529. MR 0420735
- Peter Schneider and Jeremy Teitelbaum, Algebras of $p$-adic distributions and admissible representations, Invent. Math. 153 (2003), no. 1, 145â196. MR 1990669, DOI https://doi.org/10.1007/s00222-002-0284-1
- The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2014.
Additional Information
Konstantin Ardakov
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
Email:
ardakov@maths.ox.ac.uk
Simon Wadsley
Affiliation:
Homerton College, Cambridge, CB2 8PH, United Kingdom
MR Author ID:
770243
Email:
S.J.Wadsley@dpmms.cam.ac.uk
Received by editor(s):
April 27, 2016
Received by editor(s) in revised form:
June 9, 2017
Published electronically:
July 19, 2018
Additional Notes:
The first author was supported by EPSRC grant EP/L005190/1.
Article copyright:
© Copyright 2018
University Press, Inc.