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Variants of formal nearby cycles

Part of: Arithmetic problems. Diophantine geometry (Co)homology theory

Published online by Cambridge University Press:  19 August 2013

Yoichi Mieda
Yoichi Mieda*
Affiliation:
The Hakubi Center for Advanced Research/Department of Mathematics, Kyoto University, Kyoto, 606–8502, Japan ([email protected])
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Abstract

In this paper, we introduce variants of formal nearby cycles for a locally noetherian formal scheme over a complete discrete valuation ring. If the formal scheme is locally algebraizable, then our nearby cycle gives a generalization of Berkovich’s formal nearby cycle. Our construction is entirely scheme theoretic, and does not require rigid geometry. Our theory is intended for applications to the local study of the cohomology of Rapoport–Zink spaces.

Keywords

nearby cycleétale cohomologyrigid geometry

MSC classification

Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 14G20: Local ground fields 14G22: Rigid analytic geometry
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 4 , October 2014 , pp. 701 - 752
Copyright
©Cambridge University Press 2013 

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References

Berkovich, V. G., Vanishing cycles for formal schemes, Invent. Math. 115 (3) (1994), 539–571.CrossRefGoogle Scholar
Berkovich, V. G., Vanishing cycles for formal schemes. II, Invent. Math. 125 (2) (1996), 367–390.CrossRefGoogle Scholar
Bosch, S. and Lütkebohmert, W., Formal and rigid geometry. I. Rigid spaces, Math. Ann. 295 (2) (1993), 291–317.CrossRefGoogle Scholar
Conrad, B., Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (3) (2007), 205–257.Google Scholar
Deligne, P., Théorie de Hodge. III, Publ. Math. Inst. Hautes Études Sci. (44) (1974), 5–77.CrossRefGoogle Scholar
Deligne, P., Cohomologie étale, Lecture Notes in Mathematics, Volume 569 (Springer-Verlag, Berlin, 1977).CrossRefGoogle Scholar
Ekedahl, T., On the adic formalism, in The Grothendieck Festschrift, Vol. II, Progress in Mathematics, Volume 87, pp. 197–218 (Birkhäuser Boston, Boston, MA, 1990).Google Scholar
Fargues, L., Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales, Astérisque (291) (2004), 1–199 Variétés de Shimura, espaces de Rapoport–Zink et correspondances de Langlands locales.Google Scholar
Fujiwara, K., Theory of tubular neighborhood in étale topology, Duke Math. J. 80 (1) (1995), 15–57.CrossRefGoogle Scholar
Gabber, O., Finiteness theorems for étale cohomology of excellent schemes, Notes of the Exposition in the Conference on the Occasion of the Sixty First Birthday of Pierre Deligne, 2005.Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique, Publ. Math. Inst. Hautes Études Sci. (4, 8, 11, 17, 20, 24, 28, 32)(1961–1967).Google Scholar
Grothendieck, A. and Dieudonné, J., Eléments de Géométrie algébrique I, in Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 166. (Springer-Verlag, Berlin, Heidelberg, New York, 1971).Google Scholar
Harris, M., On the local Langlands correspondence, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 583–597 (Higher Ed. Press, 2002).Google Scholar
Harris, M. and Taylor, R., The geometry and cohomology of some simple Shimura varieties (with an appendix by Vladimir G. Berkovich), Annals of Mathematics Studies, Volume 151 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Huber, R., A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (4) (1994), 513–551.CrossRefGoogle Scholar
Huber, R., Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, Volume E30 (Friedr. Vieweg & Sohn, Braunschweig, 1996).CrossRefGoogle Scholar
Huber, R., A comparison theorem for l-adic cohomology, Compositio Math. 112 (2) (1998), 217–235.CrossRefGoogle Scholar
Huber, R., A finiteness result for direct image sheaves on the étale site of rigid analytic varieties, J. Algebraic Geom. 7 (2) (1998), 359–403.Google Scholar
Huber, R., A finiteness result for the compactly supported cohomology of rigid analytic varieties, J. Algebraic Geom. 7 (2) (1998), 313–357.Google Scholar
Illusie, L., Autour du théorème de monodromie locale, Astérisque (223) (1994), 9–57 Périodes p-adiques (Bures-sur-Yvette, 1988).Google Scholar
Illusie, L., Laszlo, Y. and Orgogozo, F., Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l’école polytechnique 2006–2008, preprint, (http://www.math.polytechnique.fr/~orgogozo/travaux_de_Gabber/), 2012.Google Scholar
Ito, T. and Mieda, Y., Cuspidal representations in the $\ell $-adic cohomology of the Rapoport–Zink space for $\mathrm{GSp} (4)$, preprint, (arXiv:1005.5619), 2010.Google Scholar
Lütkebohmert, W., On compactification of schemes, Manuscripta Math. 80 (1) (1993), 95–111.CrossRefGoogle Scholar
Mieda, Y., On $\ell $-independence for the étale cohomology of rigid spaces over local fields, Compositio Math. 143 (2) (2007), 393–422.CrossRefGoogle Scholar
Mieda, Y., Non-cuspidality outside the middle degree of $\ell $-adic cohomology of the Lubin–Tate tower, Adv. Math. 225 (4) (2010), 2287–2297.CrossRefGoogle Scholar
Nagata, M., Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2 (1962), 1–10.Google Scholar
Nagata, M., A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ. 3 (1963), 89–102.Google Scholar
Rapoport, M., Non-Archimedean period domains, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), pp. 423–434 (Birkhäuser, 1995).CrossRefGoogle Scholar
Saito, M., Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (6) (1989), 849–995.CrossRefGoogle Scholar
Valabrega, P., On the excellent property for power series rings over polynomial rings, J. Math. Kyoto Univ. 15 (2) (1975), 387–395.Google Scholar
Valabrega, P., A few theorems on completion of excellent rings, Nagoya Math. J. 61 (1976), 127–133.CrossRefGoogle Scholar
Théorie des topos et cohomologie étale des schémas (SGA4), Lecture Notes in Mathematics, Volume 269, 270, 305 (Springer-Verlag, Berlin, 1972–1973).Google Scholar
Groupes de monodromie en géométrie algébrique (SGA7), Lecture Notes in Mathematics, Volume 288, 340, (Springer-Verlag, Berlin, 1972).Google Scholar