Shtukas with one leg II

Abstract

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).