7. On Nonarchimedean Banach Fields

  • Kiran S. Kedlaya Dept. of Mathematics, University of California, San Diego, 9500 Gilman Drive 0112, La Jolla, CA 92093, USA
Keywords: nonarchimedean Banach rings, perfectoid fields


We study the problem of whether a commutative nonarchimedean Banach ring which is algebraically a field can be topologized by a multiplicative norm. This can fail in general, but it holds for uniform Banach rings under some mild extra conditions. Notably, any perfectoid ring whose underlying ring is a field is a perfectoid field.


J. Ax, Zeros of polynomials over local fields--the Galois action,
J. Alg. 15 (1970), 417--428.
DOI 10.1016/0021-8693(70)90069-4;
zbl 0216.04703;

V. Berkovich,
Spectral Theory and Analytic Geometry over Non-Archimedean Fields,
Surveys and Monographs 33, Amer. Math. Soc., Providence, 1990.
zbl 0715.14013;

B. Bhatt, M. Morrow, and P. Scholze, Integral $p$-adic Hodge theory,
arXiv:1602.03148v1 (2016).
zbl 1046.11085;

S. Bosch, U. G\"untzer, and R. Remmert,
Non-Archimedean Analysis,
Grundlehren der Math. Wiss. 261, Springer-Verlag, Berlin, 1984.
zbl 0539.14017

S. Bosch and W. L\"utkebohmert, Formal and rigid geometry, II: Flattening techniques,
Math. Ann. 296 (1993), 403--429.
DOI 10.1007/BF01445112;
zbl 0808.14018;

A.J. Engler and A. Prestel, Valued Fields, Springer-Verlag,
Berlin, 2005.
zbl 1128.12009;

A. Escassut,
Propri\'et\'es spectrales en analyse non archim\'edienne,
Ast\'erisque 24 (1975), 157--167.

J.-M. Fontaine,
Perfecto\"ides, presque puret\'e et monodromie-poids (d'apr\`es Peter Scholze),
S\'eminaire Bourbaki, volume 2011/2012,
Ast\'erisque 352 (2013).
zbl 1325.14033;

K. Fujiwara, O. Gabber, and F. Kato, On Hausdorff completions of commutative rings in rigid geometry,
J. Algebra 332 (2011), 293--321.
DOI 10.1016/j.jalgebra.2011.02.001;
zbl 1230.13021;

O. Gabber and L. Ramero, Foundations for almost ring theory -- Release 6.95,
arXiv:math/0409584v11 (2016).
DOI 10.1007/b10047;
zbl 1045.13002;

B. Guennebaud, Sur une notion de spectre pour les alg\'ebres norm\'ees ultram\'etriques,
PhD thesis, Universit\'e de Poitiers, 1973; available at \url{http://www.ihes.fr/~gabber/GUENN.pdf}
(retrieved April 2017).

K.S. Kedlaya, $p$-adic Differential Equations,
Cambridge Univ. Press, Cambridge, 2010.

K.S. Kedlaya, Sheaves, shtukas, and stacks, lecture notes from Arizona Winter School 2017: Perfectoid spaces, available at \url{http://swc.math.arizona.edu/aws/2017/2017KedlayaNotes.pdf}.

T. Mihara, On Tate acyclicity and uniformity of Berkovich spectra and adic spectra,
Israel J. Math. 216 (2016), 61--105.
DOI 10.1007/s11856-016-1404-8;
zbl 1375.14091;
arxiv 1403.7856

K.S. Kedlaya and R. Liu, Relative $p$-adic Hodge theory: Foundations,
Ast\'erisque 371 (2015).
zbl 1370.14025;

K.S. Kedlaya and R. Liu, Relative $p$-adic Hodge theory, II: Imperfect period rings,
arXiv:1602.06899v2 (2016).

P. Schneider, Nonarchimedean Functional Analysis, Springer-Verlag, Berlin, 2002.
zbl 0998.46044;

P. Scholze, Perfectoid spaces,
Publ. Math. IH\'ES 116 (2012), 245--313.
DOI 10.1007/s10240-012-0042-x;
zbl 1263.14022;
arxiv 1111.4914
Unassigned Articles