*Peter Scholze and Jared Weinstein*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0012
- Subject:
- Mathematics, Geometry / Topology

This chapter analyzes shtukas with one leg over a geometric point in detail, and discusses the relation to (integral) p-adic Hodge theory. It focuses on the connection between shtukas with one leg ...
More

This chapter analyzes shtukas with one leg over a geometric point in detail, and discusses the relation to (integral) p-adic Hodge theory. It focuses on the connection between shtukas with one leg and p-divisible groups, and recovers a result of Fargues which states that p-divisible groups are equivalent to one-legged shtukas of a certain kind. In fact this is a special case of a much more general connection between shtukas with one leg and proper smooth (formal) schemes. Throughout, the goal is to fix an algebraically closed nonarchimedean field. The chapter then provides an overview of shtukas with one leg and p-divisible groups.Less

This chapter analyzes shtukas with one leg over a geometric point in detail, and discusses the relation to (integral) *p*-adic Hodge theory. It focuses on the connection between shtukas with one leg and *p*-divisible groups, and recovers a result of Fargues which states that *p*-divisible groups are equivalent to one-legged shtukas of a certain kind. In fact this is a special case of a much more general connection between shtukas with one leg and proper smooth (formal) schemes. Throughout, the goal is to fix an algebraically closed nonarchimedean field. The chapter then provides an overview of shtukas with one leg and *p*-divisible groups.

*Peter Scholze and Jared Weinstein*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0013
- Subject:
- Mathematics, Geometry / Topology

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 ...
More

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).Less

This chapter offers a second lecture on one-legged shtukas. It shows that a shtuka over Spa *C*^{b}, a priori defined over *Y*_{[0,INFINITY)} = Spa A_{inf} REVERSE SOLIDUS {*x*_{k}, *x*_{L}}, actually extends to *Y* = Spa A_{inf} REVERSE SOLIDUS {*x*_{k}}. 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).

*Peter Scholze and Jared Weinstein*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0014
- Subject:
- Mathematics, Geometry / Topology

This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following ...
More

This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following result, where Y = Spa Ainf REVERSE SOLIDUS {xk}. Theorem 14.2.1 posits that there is an equivalence of categories between finite free Ainf-modules and vector bundles on Y. One should think of this as being an analogue of a classical result: If (R, m) is a 2-dimensional regular local ring, then finite free R-modules are equivalent to vector bundles on (Spec R)REVERSE SOLIDUS {m}. The chapter then provides a proof of Theorem 14.2.1.Less

This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following result, where *Y* = Spa A_{inf} REVERSE SOLIDUS {*x*_{k}}. Theorem 14.2.1 posits that there is an equivalence of categories between finite free A_{inf}-modules and vector bundles on *Y*. One should think of this as being an analogue of a classical result: If (*R*, m) is a 2-dimensional regular local ring, then finite free *R*-modules are equivalent to vector bundles on (Spec *R*)REVERSE SOLIDUS {m}. The chapter then provides a proof of Theorem 14.2.1.