nLab perfectoid space

Idea

Perfectoid spaces are a variant of Huber spaces in analytic geometry. The concept was introduced (Scholze 11) in order to generalize the classical theorem of (Fontaine-Winterberger 79) (see also at function field analogy).

This theorem establishes an isomorphism between the absolute Galois groups of the extension $L \coloneqq \cup_{n}\mathbb{Q}_{p}[p^{1/p^{n}}]$ of the p-adic numbers and of the perfection $L^{\flat} \coloneqq \cup_{n}\mathbb{F}_{p}((t^{1/p^{n}}))$ of the field of Laurent series of the finite field $\mathbb{F}_p$ (the isomorphism between Galois groups continues to hold after passing to the completions of these fields - it is these completions that are more properly called perfectoid). The field $L^{\flat}$ is called the tilt of $L$. In (Scholze 11) this is generalized by the statement that the perfectoid spaces over fields related this way are equivalent. (See Bhatt 14 for a review).

Definitions

The reference for these definitions are Weinstein17.

Perfectoid fields

• A nonarchimedean field $K$ of residue characteristic $p$ is perfectoid if its value group is nondiscrete and the $p$-th power Frobenius map on $K^{\circ}/p$ is surjective (here $K^{\circ}$ is the subset of power bounded elements).

Perfectoid rings

• A Huber ring is a topological ring $A$ which contains an open subring $A_{0}$ which is adic with respect to a finitely generated ideal of definition $I\subseteq A_{0}$.

• A Huber ring $A$ is Tate if it contains a topologically nilpotent unit $\varpi$ (also called a pseudouniformizer).

• A Tate Huber ring is perfectoid if it is complete, uniform (the subset of power-bounded elements is bounded), and contains a pseudo-uniformizer $\varpi$ such that $\varpi^{p}\vert p$ and the p-th power map $A/\varpi\to A/\varpi^{p}$ is an isomorphism.

Perfectoid space

• A perfectoid space is an adic space which can be covered by affinoids $Spa(A,A^{+})$, where $A$ is a perfectoid ring.

Properties

Tilting

The concept of tilting allows us to generalize the theorem of Fontaine and Wintenberger on the isomorphism of the absolute Galois groups of a perfectoid field and its tilt mentioned above, to perfectoid rings, and further to perfectoid spaces.

Definition

(ScholzeWeinstein20, Definition 6.2.1) Let $R$ be a perfectoid ring. The tilt of $R$ is

$$R^{\flat}=\underset {\underset{x\mapsto x^{p}}{\leftarrow}} {\lim}R$$

A priori this is a topological multiplicative monoid, so turn it into a topological ring we equip it with the addition structure given by

$$(x+y)^{(i)}=\lim_{n\to\infty}(x^{(i+n)}+y^{(i+n)})^{p^{n}}.$$

Proposition

(ScholzeWeinstein20, Theorem 7.4.5)

Let $R$ be a perfectoid ring and let $R^{\flat}$ be its tilt.

• For any finite etale $R$-algebra $S$, $S$ is perfectoid.

• Tilting $S\mapsto S^{\flat}$ induces an equivalence between finite etale $R$-algebras and finite etale $R^{\flat}$-algebras.

• For any finite etale $R$-algebra $S$, the algebra $S^{\circ}$, is almost finite etale over $R^{\circ}$.

The tilting construction “glues” and therefore carries over from perfectoid rings to perfectoid spaces; in other words to a perfectoid space $X$ we can take the tilt to obtain a perfectoid space $X^{\flat}$ of characteristic $p$ (ScholzeWeinstein20, 7.1).

Proposition

(ScholzeWeinstein20, Corollary 7.5.3)

There exists an etale site $X_{et}$ such that $X_{et}\cong X_{et}^{\flat}$ and $H^{i}(X_{et},\mathcal{O}_{X}^{+})$ is almost zero for all $i\geq 1$ and for all affinoids $X$.

Generalizations

The concept of perfectoid space can be generalized into that of a diamond, which is a quotient of a perfectoid space of characteristic $p$ by a perfectoid equivalence relation (ScholzeWeinstein20, Definition 8.3.1). The concept of diamond can similarly be generalized into that of a v-sheaf, and in particular a small v-sheaf is a quotient of a diamond by a diamond equivalence relation (ScholzeWeinstein20, Proposition 17.2.2).

Applications

One application of perfectoid spaces is in relating Galois representations to torsion in the cohomology of Shimura varieties. In turn, using the excision sequence, one can relate this to the cohomology of arithmetic manifolds that are not Shimura varieties (for example Bianchi manifolds, which are quotients of hyperbolic 3-space by an arithmetic subgroup). This is surveyed in section 5 of Weinstein15. Subsequent work in progress by Ana Caraiani and James Newton (see Caraiani’s talk at IHES, CaraianiIHES) make use of this to prove a version of the modularity theorem for elliptic curves over quadratic imaginary fields.

Related concepts

• Berkovich space

• global analytic geometry

• p-adic Hodge theory

• perfectoid field

• prismatic cohomology

References

Exposition includes

• Michael Harris, The perfectoid concept: Test case for an absent theory (pdf)

The concept is due to

• Peter Scholze, Perfectoid spaces (arXiv:1111.4914)

motivated by

• Jean-Marc Fontaine, Jean-Pierre Wintenberger, Extensions algébrique et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris Sér. A–B 288(8) (1979), A441–A444

Review includes

• Bhargav Bhatt, What is… a perfectoid space?, Notices of the AMS, volume 61, number 9 (pdf)

• Bhargav Bhatt, Lecture notes for a class on perfectoid spaces, (pdf)

• Peter Scholze, Perfectoid spaces: a survey (arXiv:1303.5948)

• Jared Weinstein, Adic spaces, lecture notes for the 2017 Arizona Winter School pdf

Perfectoid spaces and related concepts were the topic of a course at Berkeley in 2014, whose lecture notes have now been made into a book:

• Peter Scholze and Jared Weinstein, Berkeley Lectures on p-adic Geometry pdf

Some applications of perfectoid spaces are discussed in

• Jared Weinstein, Reciprocity Laws and Galois Representations: Recent Breakthroughs pdf

Progress in the theory is being applied to prove a modularity theorem for elliptic curves over quadratic imaginary fields, discussed in

• Ana Caraiani, Modularity over CM Fields, (Talk at IHES YouTube)

See also

• Wikipedia, Perfectoid space

Formalization in type theory (in Lean):

• Kevin Buzzard, Johan Commelin, Patrick Massot, lean-perfectoid-spaces