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つづき

An important consequence of these results is Corollary 3.1, which provides a function from
a suitable Fargues-Fontaine curve to the isomorphism class of the tempered fundamental group
of a fixed variety (as above) which provides a natural way of labeling the copies obtained here
by closed points of a suitable Fargues-Fontaine curve.
In the last section of the paper I show that there is an entirely analogous theory of untilts of
topological fundamental groups of connected Riemann surfaces.
This note began as a part of another note, [Jos20a], which I put into a limited circulation some time in July 2020, outlining my own approach to some constructions of [Moc12a;Moc12b; Moc12c; Moc12d]. Peter Scholze immediately, but gently, pointed out that the section of [Jos20a], from which the present note is extracted, needed some details.
At that time I was readying another note, [Jos20b], for wider circulation and addressing the issue noted by
Scholze took longer and on the way I was able to substantially strengthen and clarify my results
(which appear here). So ultimately I decided that it would be best to publish the present note
separately (while preparation of [Jos20a] continued). My thanks are due to Peter Scholze, and
also to Yuichiro Hoshi, Emmanuel Lepage, and Jacob Stix, for promptly providing comments,
suggestions or corrections.

2 The main theorem

Lemma 2.1. Let K be a valued field and let R ⊂ K be the valuation ring. The following
conditions are equivalent:
(1) K is an algebraically closed field, complete with respect to a rank one non-archimedean
valuation and with residue characteristic p > 0.
(2) K is an algebraically closed, perfectoid field.
Proof. A perfectoid field has residue characteristic p > 0 and is complete with respect to a rank one valuation.
(引用終り)
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