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Perfectoid spaces and the weight-monodromy conjecture, after Peter Scholze

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Abstract

The monodromy weight conjecture is one of the main remaining open problems on Galois representations. It implies that the local Galois action on the ℓ-adic cohomology of a proper smooth variety is almost completely determined by the traces. Peter Scholze proved the conjecture in many cases including smooth complete intersections in a projective space, using a new powerful tool in rigid geometry called perfectoid spaces. The main arguments of the proof as well as basic ingredients in the theory of perfectoid spaces are presented.

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References

  1. Deligne, P.: La conjecture de Weil. II. Publ. Math. Inst. Hautes Études Sci. 52, 137–252 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Faltings, G.: Almost étale extensions. Cohomologies p-adiques et applications arithmétiques, II. Astérisque 279, 185–270 (2002)

    MathSciNet  Google Scholar 

  3. Fontaine, J.-M.: Perfectoïdes, presque pureté et monodromie-poids, d’après Peter Scholze. Séminaire Bourbaki, 64ème année, pp. 2011–2012, no 1057

  4. Fontaine, J.-M., Wintenberger, J.-P.: Extensions algébrique et corps des normes des extensions APF des corps locaux. C. R. Acad. Sci. Paris Sér. A-B 288, 441–444 (1979)

    MATH  MathSciNet  Google Scholar 

  5. Gabber, O., Ramero, L.: Almost Ring Theory. Lect. Notes Math., vol. 1800. Springer, Berlin (2003)

    MATH  Google Scholar 

  6. Huber, R.: Étale cohomology of rigid analytic varieties and adic spaces. Aspects Math. E, vol. E30. (1996)

    Book  MATH  Google Scholar 

  7. Illusie, L.: Complexe cotangent et déformations I. Lect. Notes Math., vol. 239. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  8. Scholze, P.: Perfectoid spaces. Publ. Math. Inst. Hautes Études Sci. 116, 245–313 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Tsuji, T.: Notes on almost étale extensions of Faltings. Preprint

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Acknowledgements

The author thanks A. Abbes, T. Mihara, K. Miyatani, K. Tokimoto and N. Umezaki for useful comments on a preliminary version.

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  1. Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Takeshi Saito

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  1. Takeshi Saito
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Correspondence to Takeshi Saito.

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Saito, T. Perfectoid spaces and the weight-monodromy conjecture, after Peter Scholze. Acta Math Vietnam. 39, 55–68 (2014). https://doi.org/10.1007/s40306-013-0044-x

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  • DOI: https://doi.org/10.1007/s40306-013-0044-x

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