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Berkeley Lectures on p-adic Geometry(AMS-207)$
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Peter Scholze and Jared Weinstein

Print publication date: 2020

Print ISBN-13: 9780691202082

Published to Princeton Scholarship Online: January 2021

DOI: 10.23943/princeton/9780691202082.001.0001

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Shtukas with one leg II

Shtukas with one leg II

(p.108) Lecture 13 Shtukas with one leg II
Berkeley Lectures on p-adic Geometry

Peter Scholze

Jared Weinstein

Princeton University Press

This chapter offers a second lecture on one-legged shtukas. It shows that a shtuka over Spa Cb, a priori defined over Y[0,INFINITY) = Spa Ainf REVERSE SOLIDUS {xk, xL}, actually extends to Y = Spa Ainf REVERSE SOLIDUS {xk}. In doing so, the chapter considers the theory of φ‎-modules over the Robba ring, due to Kedlaya. These are in correspondence with vector bundles over the Fargues-Fontaine curve. The chapter then looks at the proposition that the space Y is an adic space. It also sketches a proof that the functor described in Theorem 13.2.1 is fully faithful. This is more general, and works if C is any perfectoid field (not necessarily algebraically closed).

Keywords:   one-legged shtukas, shtukas, φ-modules, Robba ring, Kedlaya, vector bundles, Fargues-Fontaine curve, adic space, perfectoid field

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