Paper · Sep 21, 2017

Etale cohomology of diamonds

Peter Scholze

math.AGarXiv:1709.07343v4
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Abstract

Motivated by problems on the étale cohomology of Rapoport–Zink spaces and their generalizations, as well as Fargues's geometrization conjecture for the local Langlands correspondence, we develop a six functor formalism for the étale cohomology of diamonds, and more generally small v-stacks on the category of perfectoid spaces of characteristic . Using a natural functor from analytic adic spaces over to diamonds which identifies étale sites, this induces a similar formalism in that setting, which in the noetherian setting recovers the formalism from Huber's book.

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Submitted
Sep 21, 2017
Last revised
Apr 14, 2026 (v4)
Subjects
math.AG, math.NT

Comments: 168 pages, final version to appear in Asterisque

Geometrization

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Geometrization of the local Langlands correspondence: an overview (Laurent Fargues, 2016)2Fargues 2016 Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local (Laurent Fargues, 2017)1Fargues 2017 Higgs bundles and geometric structures on manifolds (Daniele Alessandrini, 2018)6Alessandrini 2018 Uniform models and short curves for random 3-manifolds (Peter Feller, Alessandro Sisto and Gabriele Viaggi, 2019)8Feller 2019 Canonical geometrization of orientable 3-manifolds defined by vector-colourings of 3-polytopes (Nikolai Erokhovets, 2020)3Erokhovets 2020 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)5de Freitas 2022 Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes (Tanay Gupta, Sukanta Panda and Rajib Saha, 2026)7Gupta 2026 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)4Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local

    Laurent Fargues

    Topics Geometrization · Shared terms “Langlands”, “geometrization conjecture”, “correspondence”

  2. Geometrization of the local Langlands correspondence: an overview

    Laurent Fargues

    Topics Geometrization · Shared terms “Langlands correspondence”, “local Langlands”, “geometrization conjecture”

  3. Canonical geometrization of orientable -manifolds defined by vector-colourings of -polytopes

    Nikolai Erokhovets

    Topics Geometrization · Shared terms “geometrization conjecture”

  4. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al.

    Topics Geometrization · Shared terms “geometrization conjecture”

  5. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

    Topics Geometrization

  6. Higgs bundles and geometric structures on manifolds

    Daniele Alessandrini

    Topics Geometrization · Shared terms “geometrization conjecture”

  7. Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes

    Tanay Gupta, Sukanta Panda and Rajib Saha

    Topics Geometrization · Shared terms “formalism”, “geometrization conjecture”, “Motivated”

  8. Uniform models and short curves for random 3-manifolds

    Peter Feller, Alessandro Sisto and Gabriele Viaggi

    Topics Geometrization · Shared terms “geometrization conjecture”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982)

In the guidebook

  1. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.