*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/9780691202082.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic ...
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This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.Less

This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.

*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/9780691202082.003.0018
- Subject:
- Mathematics, Geometry / Topology

This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also ...
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This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also certain non-analytic objects as v-sheaves. The chapter first analyzes the behavior on topological spaces. Let X be any pre-adic space over Zp. This is not a diamond, but the chapter shows that it is a v-sheaf. It assesses some properties of this construction. The chapter then looks at applications to local models and integral models of Rapoport-Zink spaces. By passage to the maximal unramified extension and Galois descent, one can assume that k is algebraically closed.Less

This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also certain non-analytic objects as v-sheaves. The chapter first analyzes the behavior on topological spaces. Let *X* be any pre-adic space over **Z**p. This is not a diamond, but the chapter shows that it is a v-sheaf. It assesses some properties of this construction. The chapter then looks at applications to local models and integral models of Rapoport-Zink spaces. By passage to the maximal unramified extension and Galois descent, one can assume that *k* is 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/9780691202082.003.0012
- 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 ...
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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.