Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Comparison results for étale cohomology in rigid geometry


Author: Yoichi Mieda
Journal: J. Algebraic Geom. 23 (2014), 91-115
DOI: https://doi.org/10.1090/S1056-3911-2013-00607-4
Published electronically: September 9, 2013
MathSciNet review: 3121849
Full-text PDF

Abstract | References | Additional Information

Abstract: We give some comparison results between étale cohomology of schemes and that of associated rigid spaces. First we work in the framework of Fujiwara spaces, and prove a comparison theorem for the derived direct image of morphisms which are not necessarily proper. We need the assumption that the base of the comparison functor consists of schemes of finite type over a field. Next, we consider an analogous problem for adic spaces. In this case, we can prove the comparison theorem under the condition that the base of the comparison functor consists of schemes of finite type over a complete discrete valuation ring with equal characteristic. Finally we propose a definition of the nearby cycle functor for an adic space over a complete discrete valuation ring and give a comparison result on it.


References [Enhancements On Off] (What's this?)

  • [Ber96a] Vladimir G. Berkovich, Vanishing cycles for formal schemes. II, Invent. Math. 125 (1996), no. 2, 367-390. MR 1395723 (98k:14031), https://doi.org/10.1007/s002220050078
  • [Ber96b] Vladimir G. Berkovich, Vanishing cycles for non-Archimedean analytic spaces, J. Amer. Math. Soc. 9 (1996), no. 4, 1187-1209. MR 1376692 (97e:32037), https://doi.org/10.1090/S0894-0347-96-00214-7
  • [dJ96] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51-93. MR 1423020 (98e:14011)
  • [Fuj95] Kazuhiro Fujiwara, Theory of tubular neighborhood in étale topology, Duke Math. J. 80 (1995), no. 1, 15-57. MR 1360610 (97d:14028), https://doi.org/10.1215/S0012-7094-95-08002-8
  • [Gab94] Ofer Gabber, Affine analog of the proper base change theorem, Israel J. Math. 87 (1994), no. 1-3, 325-335. MR 1286833 (95f:14035), https://doi.org/10.1007/BF02773001
  • [Hub93] R. Huber, Continuous valuations, Math. Z. 212 (1993), no. 3, 455-477. MR 1207303 (94e:13041), https://doi.org/10.1007/BF02571668
  • [Hub94] R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, 513-551. MR 1306024 (95k:14001), https://doi.org/10.1007/BF02571959
  • [Hub96] Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1734903 (2001c:14046)
  • [HV08] Urs Hartl and Eva Viehmann, The Newton stratification on deformations of local $ G$-shtukas, J. Reine Angew. Math. 656 (2011), 87-129. MR 2818857, https://doi.org/10.1515/CRELLE.2011.044
  • [Ill04] Luc Illusie, On semistable reduction and the calculation of nearby cycles, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 785-803. MR 2099086 (2005i:14022)
  • [IM10] T. Ito and Y. Mieda, Cuspidal representations in the $ \ell $-adic cohomology of the Rapoport-Zink space for $ \mathrm {GSp}(4)$, preprint, arXiv:1005.5619, 2010.
  • [Mie10a] Yoichi Mieda, Non-cuspidality outside the middle degree of $ \ell $-adic cohomology of the Lubin-Tate tower, Adv. Math. 225 (2010), no. 4, 2287-2297. MR 2680204 (2011g:22029), https://doi.org/10.1016/j.aim.2010.04.021
  • [Mie10b] Yoichi Mieda, Variants of formal nearby cycles, preprint, arXiv:1005.5616, 2010.
  • [RG71] Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de ``platification'' d'un module, Invent. Math. 13 (1971), 1-89 (French). MR 0308104 (46 #7219)
  • [EGA] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. (1961-1967), no. 4, 8, 11, 17, 20, 24, 28, 32.
  • [SGA4] Théorie des topos et cohomologie étale des schémas (SGA4), Lecture Notes in Mathematics, Vol. 269, 270, 305, Springer-Verlag, Berlin, 1972-1973.


Additional Information

Yoichi Mieda
Affiliation: Department of Mathematics, The Hakubi Center for Advanced Research, Kyoto University, Kyoto, 606-8502, Japan
Email: mieda@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S1056-3911-2013-00607-4
Received by editor(s): January 19, 2011
Received by editor(s) in revised form: June 6, 2011
Published electronically: September 9, 2013
Article copyright: © Copyright 2013 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society