Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves

Kedlaya, Kiran; Temkin, Michael

doi:10.1090/proc/13818

Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves
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by Kiran S. Kedlaya and Michael Temkin PDF

Proc. Amer. Math. Soc. 146 (2018), 489-495

Abstract:

We show that for $k$ a perfect field of characteristic $p$, there exist endomorphisms of the completed algebraic closure of $k((t))$ which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for $p$ a prime and $\mathbb {C}_p$ a completed algebraic closure of $\mathbb {Q}_p$, there exist closed points of the Fargues-Fontaine curve associated to $\mathbb {C}_p$ whose residue fields are not (even abstractly) isomorphic to $\mathbb {C}_p$ as topological fields.

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