Perfectoid rings

This chapter examines perfectoid spaces. A Huber ring R is Tate if it contains a topologically nilpotent unit; such elements are called pseudo-uniformizers. One can more generally define when an analytic Huber ring is perfectoid. There are also notions of integral perfectoid rings which are not analytic. In this course, the perfectoid rings are all Tate. It would have been possible to proceed with the more general definition of perfectoid ring as a kind of analytic Huber ring. However, being analytic is critical for the purposes of the course. The chapter then looks at tilting and sousperfectoid rings. The class of sousperfectoid rings has good stability properties.

Examples of diamonds

This chapter assesses some interesting examples of diamonds. So far, the only example encountered is the self-product of copies of Spd Qp. The chapter first studies this self-product. It is useful to keep in mind that a diamond can have multiple “incarnations.” Another important class of diamonds, which in fact were one of the primary motivations for their definition, is the category of Banach-Colmez spaces. Recently, le Bras has reworked their theory in terms of perfectoid spaces. The category of Banach-Colmez spaces over C is the thick abelian subcategory of the category of pro-étale sheaves of Qp-modules. This is similar to a category considered by Milne in characteristic p.

Diamonds

This chapter investigates the notion of a diamond. The idea is that there should be a functor which “forgets the structure morphism to Zp.” The desired quotient in the example provided in the chapter exists in a category of sheaves on the site of perfectoid spaces with pro-étale covers. The chapter then defines pro-étale morphisms between perfectoid spaces. A morphism of perfectoid spaces is pro-étale if it is locally (on the source and target) affinoid pro-étale. The intuitive definition of diamonds involved the tilting functor in case of perfectoid spaces of characteristic 0. For this reason, diamonds are defined as certain pro-étale sheaves on the category of perfectoid spaces of characteristic p.

Introduction

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

Mixed-characteristic shtukas

This chapter looks at mixed-characteristic shtukas. Much of the theory of mixed-characteristic shtukas is motivated by the structures appearing in (integral) p-adic Hodge theory. The chapter assesses Drinfeld's shtukas and local shtukas. In the mixed characteristic setting, X will be replaced with Spa Zp. The test objects S will be drawn from Perf, the category of perfectoid spaces in characteristic p. For an object, a shtuka over S should be a vector bundle over an adic space, together with a Frobenius structure. The product is not meant to be taken literally (if so, one would just recover S), but rather it is to be interpreted as a fiber product over a deeper base. Motivated by this, the chapter then defines an analytic adic space and shows that its associated diamond is the appropriate product of sheaves on Perf.

Diamonds II

This chapter evaluates the complements on the pro-étale topology. It addresses two issues raised in the previous lecture on the pro-étale topology. The first issue concerned descent, or more specifically pro-étale descent for perfectoid spaces. The other issue was that the property of being a pro-étale morphism is not local for the pro-étale topology on the target. The chapter then looks at quasi-pro-étale morphisms, as well as G-torsors. A morphism of perfectoid spaces is quasi-pro-étale if for any strictly totally disconnected perfectoid space with a map, the pullback is pro-étale. Using this definition, one can give an equivalent characterization of diamonds.

Perfectoid spaces

This chapter offers a second lecture on perfectoid spaces. A perfectoid Tate ring R is a complete, uniform Tate ring containing a pseudo-uniformizer. A perfectoid space is an adic space covered by affinoid adic spaces with R perfectoid. The term “affinoid perfectoid space” is ambiguous. The chapter then looks at the tilting process and the tilting equivalence. The tilting equivalence extends to the étale site of a perfectoid space. Why is it important to study perfectoid spaces? The chapter puts forward a certain philosophy which indicates that perfectoid spaces may arise even when one is only interested in classical objects.

Berkeley Lectures on p-adic Geometry

This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained.