Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

See all formats and pricing
More options …
Volume 2018, Issue 740


Stably uniform affinoids are sheafy

Kevin Buzzard / Alain Verberkmoes
Published Online: 2016-01-29 | DOI: https://doi.org/10.1515/crelle-2015-0089


We develop some of the foundations of affinoid pre-adic spaces without Noetherian or finiteness hypotheses. We give explicit examples of non-adic affinoid pre-adic spaces, and also a new condition ensuring that the structure presheaf on Spa(R,R+) is a sheaf. This condition can be used to give a new proof that the spectrum of a perfectoid algebra is an adic space.


  • [1]

    V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr. 33, American Mathematical Society, Providence 1990. Google Scholar

  • [2]

    S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean analysis, Grundlehren Math. Wiss. 261, Springer, Berlin 1984. Google Scholar

  • [3]

    N. Bourbaki, Éléments de mathématique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtrations et topologies. Chapitre 4: Idéaux premiers associés et décomposition primaire, Actual. Sci. Indust. 1293, Hermann, Paris 1961. Google Scholar

  • [4]

    N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris 1971. Google Scholar

  • [5]

    N. Bourbaki, Éléments de mathématique. Espaces vectoriels topologiques. Chapitres 1 à 5, Masson, Paris 1981. Google Scholar

  • [6]

    A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné). I: Le langage des schémas, Publ. Math. Inst. Hautes Études Sci. 4 (1960), 5–228. Google Scholar

  • [7]

    T. Henkel, An open mapping theorem for rings which have a zero sequence of units, preprint (2014), http://arxiv.org/abs/1407.5647v2.

  • [8]

    R. Huber, Continuous valuations, Math. Z. 212 (1993), no. 3, 455–477. CrossrefGoogle Scholar

  • [9]

    R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, 513–551. CrossrefGoogle Scholar

  • [10]

    R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects Math. E30, Friedr. Vieweg & Sohn, Braunschweig 1996. Google Scholar

  • [11]

    T. Mihara, On Tate acyclicity and uniformity of Berkovich spectra and adic spectra, preprint (2014), http://arxiv.org/abs/1403.7856v1.

  • [12]

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

  • [13]

    P. Scholze, Perfectoid spaces: A survey, Current developments in mathematics 2012 (Cambridge 2012), International Press, Somerville (2013), 193–227. Google Scholar

  • [14]

    T. Wedhorn, Adic spaces, unpublished notes (2012).

About the article

Received: 2014-05-16

Revised: 2015-09-14

Published Online: 2016-01-29

Published in Print: 2018-07-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 740, Pages 25–39, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0089.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Alberto Vezzani
Compositio Mathematica, 2019, Volume 155, Number 1, Page 38
Alberto Vezzani
Journal of Pure and Applied Algebra, 2018

Comments (0)

Please log in or register to comment.
Log in