Home Extending torsors on the punctured Spec(Ainf)
Article
Licensed
Unlicensed Requires Authentication

Extending torsors on the punctured Spec(Ainf)

  • Johannes Anschütz ORCID logo EMAIL logo
Published/Copyright: January 6, 2022

Abstract

We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine’s ring Ainf, extend to the whole spectrum. Using descent we can extend a similar result for the ring 𝔖 of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the BdR+-affine Grassmannian to the Witt vector affine flag variety.

Acknowledgements

We thank Peter Scholze heartily for his interest and suggestions regarding this topic, especially for his suggestion to consider Mal’cev–Neumann series. Moreover, we heartily thank Kęstutis Česnavičius, Ian Gleason, Daniel Kirch, João Lourenço and Sebastian Posur for discussions and comments on this paper.

References

[1] J. Alper, Adequate moduli spaces and geometrically reductive group schemes, preprint (2010), https://arxiv.org/abs/1005.2398. 10.14231/AG-2014-022Search in Google Scholar

[2] A. Beauville and Y. Laszlo, Un lemme de descente, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340. Search in Google Scholar

[3] B. Bhatt and A. Mathew, The arc-topology, preprint (2018), https://arxiv.org/abs/1807.04725. 10.1215/00127094-2020-0088Search in Google Scholar

[4] B. Bhatt, M. Morrow and P. Scholze, Integral p-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018), no. 1, 219–397. 10.1007/s10240-019-00102-zSearch in Google Scholar

[5] B. Bhatt and P. Scholze, Projectivity of the Witt vector affine Grassmannian, Invent. Math. 209 (2017), no. 2, 329–423. 10.1007/s00222-016-0710-4Search in Google Scholar

[6] B. Bhatt and P. Scholze, Prisms and prismatic cohomology, preprint (2019), https://arxiv.org/abs/1905.08229. 10.4007/annals.2022.196.3.5Search in Google Scholar

[7] P. Breutmann, Functoriality of moduli spaces of global 𝔾-shtukas, preprint (2019), https://arxiv.org/abs/1902.10602. Search in Google Scholar

[8] F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Publ. Math. Inst. Hautes Études Sci. 60 (1984), 197–376. 10.1007/BF02700560Search in Google Scholar

[9] K. Česnavičius, Purity for the Brauer group, Duke Math. J. 168 (2019), no. 8, 1461–1486. 10.1215/00127094-2018-0057Search in Google Scholar

[10] J.-L. Colliot-Thélène and J.-J. Sansuc, Fibrés quadratiques et composantes connexes réelles, Math. Ann. 244 (1979), no. 2, 105–134. 10.1007/BF01420486Search in Google Scholar

[11] O. Endler, Valuation theory, Universitext, Springer, New York 1972. 10.1007/978-3-642-65505-0Search in Google Scholar

[12] L. Fargues and J.-M. Fontaine, Courbes et fibrés vectoriels en théorie de Hodge p-adique, Astérisque 406, Société Mathématique de France, Paris 2018. 10.24033/ast.1056Search in Google Scholar

[13] K. Fujiwara, Theory of tubular neighborhood in étale topology, Duke Math. J. 80 (1995), no. 1, 15–57. 10.1215/S0012-7094-95-08002-8Search in Google Scholar

[14] O. Gabber and L. Ramero, Almost ring theory, Lecture Notes in Math. 1800, Springer, Berlin 2003. 10.1007/b10047Search in Google Scholar

[15] P. Gille, Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique 2, Compos. Math. 125 (2001), no. 3, 283–325. 10.1023/A:1002473132282Search in Google Scholar

[16] I. Gleason, Specialization maps for Scholzes category of diamonds, preprint (2020), https://arxiv.org/abs/2012.05483. Search in Google Scholar

[17] A. Grothendieck, Groupes de type multiplicatif: Homomorphismes dans un schéma en groupes, Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963/64), Inst. Hautes Études Sci., Paris (1964), Fasc. 3, Exposé 9, 37. 10.1007/BFb0059007Search in Google Scholar

[18] A. Grothendieck and J. A. Dieudonné, Éléments de géométrie algébrique. I, Grundlehren Math. Wiss. 166, Springer, Berlin 1971. Search in Google Scholar

[19] D. Hansen, Vanishing and comparison theorems in rigid analytic geometry, Compos. Math. 156 (2020), no. 2, 299–324. 10.1112/S0010437X19007371Search in Google Scholar

[20] U. Hartl and E. Viehmann, The generic fiber of moduli spaces of bounded local G-shtukas, preprint (2017), https://arxiv.org/abs/1712.07936. Search in Google Scholar

[21] R. T. Hoobler, When is Br(X)=Br(X)?, Brauer groups in ring theory and algebraic geometry (Wilrijk 1981), Lecture Notes in Math. 917, Springer, Berlin (1982), 231–244. 10.1007/BFb0092238Search in Google Scholar

[22] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects Math. E30, Friedr. Vieweg & Sohn, Braunschweig 1996. 10.1007/978-3-663-09991-8Search in Google Scholar

[23] A. B. Ivanov, On ind-representability of loop Deligne–Lusztig sheaves, preprint (2020), https://arxiv.org/abs/2003.04399. Search in Google Scholar

[24] K. S. Kedlaya, Some ring-theoretic properties of 𝐀inf, p-adic Hodge theory, Simons Symp. 1, Springer, Cham (2020), 129–141. 10.1007/978-3-030-43844-9_4Search in Google Scholar

[25] M. Kisin, Integral canonical models of Shimura varieties, J. Théor. Nombres Bordeaux 21 (2009), no. 2, 301–312. 10.5802/jtnb.672Search in Google Scholar

[26] M. Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), no. 4, 967–1012. 10.1090/S0894-0347-10-00667-3Search in Google Scholar

[27] M. Kisin and G. Pappas, Integral models of Shimura varieties with parahoric level structure, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 121–218. 10.1007/s10240-018-0100-0Search in Google Scholar

[28] E. Landvogt, A compactification of the Bruhat–Tits building, Lecture Notes in Math. 1619, Springer, Berlin 1996. 10.1007/BFb0094594Search in Google Scholar

[29] J. Lang and J. Ludwig, 𝔸inf is infinite dimensional, J. Inst. Math. Jussieu 20 (2021), no. 6, 1983–1989. 10.1017/S1474748020000201Search in Google Scholar

[30] S. Lang, On quasi algebraic closure, Ann. of Math. (2) 55 (1952), 373–390. 10.2307/1969785Search in Google Scholar

[31] M. Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet, Publ. Math. Inst. Hautes Études Sci. (1962), no. 14, 47–75. 10.1007/BF02684326Search in Google Scholar

[32] M. Lieblich, Twisted sheaves and the period-index problem, Compos. Math. 144 (2008), no. 1, 1–31. 10.1112/S0010437X07003144Search in Google Scholar

[33] J. N. P. Lourenço, Grassmanniennes affines tordues sur les entiers, preprint (2019), https://arxiv.org/abs/1912.11918. Search in Google Scholar

[34] B. Poonen, Maximally complete fields, Enseign. Math. (2) 39 (1993), no. 1–2, 87–106. Search in Google Scholar

[35] M. Rapoport and M. Richartz, On the classification and specialization of F-isocrystals with additional structure, Compos. Math. 103 (1996), no. 2, 153–181. Search in Google Scholar

[36] T. Richarz, Affine Grassmannians and geometric Satake equivalences, Int. Math. Res. Not. IMRN 2016 (2016), no. 12, 3717–3767. 10.1093/imrn/rnv226Search in Google Scholar

[37] P. Scholze, Etale cohomology of diamonds, preprint (2017), https://arxiv.org/abs/1709.07343. 10.2139/ssrn.3542189Search in Google Scholar

[38] P. Scholze and J. Weinstein, Berkeley lectures on p-adic geometry, Ann. of Math. Stud. 389, Princeton University Press, Princeton 2020. 10.23943/princeton/9780691202082.001.0001Search in Google Scholar

[39] J.-P. Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Math. 5, Springer, Berlin 1994. 10.1007/BFb0108758Search in Google Scholar

[40] J.-P. Serre, Cohomologie galoisienne: progrès et problèmes, Astérisque 227 (1995), 229–257. 10.1007/978-3-642-41978-2_34Search in Google Scholar

[41] J. Tits, Reductive groups over local fields, Automorphic forms, representations and L-functions. Part 1 ((Corvallis 1977)), Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence (1979), 29–69. 10.1090/pspum/033.1/546588Search in Google Scholar

[42] X. Zhu, Affine Grassmannians and the geometric Satake in mixed characteristic, Ann. of Math. (2) 185 (2017), no. 2, 403–492. 10.4007/annals.2017.185.2.2Search in Google Scholar

[43] X. Zhu, Coherent sheaves on the stack of langlands parameters, preprint (2020), https://arxiv.org/abs/2008.02998. Search in Google Scholar

[44] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2017. Search in Google Scholar

Received: 2021-06-11
Revised: 2021-10-05
Published Online: 2022-01-06
Published in Print: 2022-02-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2021-0077/html
Scroll to top button