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Jahresbericht der Deutschen Mathematiker-Vereinigung
Published in: Jahresbericht der Deutschen Mathematiker-Vereinigung 4/2019

30-10-2019 | Survey Article

On the Work of Peter Scholze

Author: T. Wedhorn

Published in: Jahresbericht der Deutschen Mathematiker-Vereinigung | Issue 4/2019

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Abstract

This is a survey article over some of the work of Peter Scholze. No originality is claimed.
previous article Preface
next article K. Bringmann, A. Folsom, K. Ono, L. Rolen: “Harmonic Maass Forms and Mock Modular Forms: Theory and Applications”.

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Footnotes
1
We follow here the convention of the Stacks project [56] which differs from the original definition in [4].
 
2
“Generated” means that one has to check the sheaf condition only for these special coverings.
 
3
Here we ignore some set-theoretic issues.
 
4
A separated scheme of finite type.
 
5
If \(\bar{k}\) is an algebraic closure of \(k\) and \(X(\bar{k}) = \{x \in \bar{k}^{n}; \ f_{1}(x) = \cdots = f_{r}(x) = 0\}\) for polynomials \(f_{i} \in \mathbb {F}_{q}[T_{1},\dots ,T_{n}]\), then \(F_{q}(x_{1},\dots ,x_{n}) = (x_{1}^{q},\dots ,x_{n}^{q})\) for \(x = (x_{1},\dots ,x_{n}) \in \bar{k}^{n}\).
 
6
\(\ell\)-Adic cohomology with compact support for some auxiliary prime number \(\ell \ne p\).
 
7
Needless to say that this description of the proof sweeps a whole world of technical complications under the rug. Note that by now there is also a quicker proof of the main technical point of the Fundamental Lemma using \(p\)-adic integration by Groechenig, Wyss, and Ziegler [26].
 
8
A pseudo-metric \(d\) on a set \(X\) is a distance function \(d\colon X \times X \to \mathbb {R}_{\geq 0}\) that satisfies all properties of a metric except the implication \(d(x,y) = 0 \Rightarrow x = y\).
 
9
Beware that in general the topology on \(\hat{R}\) is not the \(I\hat{R}\)-adic topology and that \(\hat{R}\) might not even by complete with respect to the \(I\hat{R}\)-adic topology.
 
10
One could also define the \(p\)-adic absolute value by \(|up^{n}|_{p} := \alpha ^{n}\) for any \(0 < \alpha < 1\) and get the same topology. The reason for choosing \(\alpha = p^{-1}\) becomes more apparent if one studies global questions involving all \(p\) simultaneously.
 
11
Although using the language of log-schemes there has been much progress in higher dimension as well.
 
12
Here we mean the ring of \(p\)-typical Witt vectors. It also exists for non-perfect rings but has to be defined differently. We will not need this generality.
 
13
“Compatible” means that \((\pi ^{1/p^{n+1}})^{p} = \pi ^{1/p^{n}}\) for all \(n\).
 
14
The analogue assertion does not hold for schemes (not even for étale morphisms of curves over a field).
 
15
With the definition of “étale” as in Definition 3.4.
 
16
One defines the notion of the étale site of a perfectoid space using Definition 3.4 as one defines the étale site as for schemes, see the introduction.
 
17
As noted above, one also has to specify a ring of integral elements. But in this case there is a canonical choice, namely \(O_{K}\langle \sigma \cap \mathbb {Z}^{n} \rangle \). A similar remark holds for the perfectoidization.
 
18
Indeed, a standard argument (e.g., [4, Rapport 3.1]) shows that \(T\frac{d}{dT}\log (\det (1 -F _{q} T)^{-1}) = \sum_{n \geq 1}\operatorname{tr}(F_{q}^{n} | H^{i}({\mathscr{X}} _{\bar{k}},\overline{\mathbb {Q}}_{\ell }))\).
 
19
It is characterized by the property that the restriction of the \(\varGamma _{K}\)-action to an open subgroup (automatically of finite index) of \(I := \operatorname{Ker}(\varGamma _{K} \to \varGamma _{k})\) is given by \(\exp (Nt_{ \ell })\), where \(t_{\ell }\colon I \to \mathbb {Z}_{\ell }(1)\) is the maximal pro-\(\ell \)-quotient, given by the limit of the homomorphism \(t_{\ell ,n}\colon I \to \mu _{\ell ^{n}}\) defined by choosing a system of \(\ell ^{n}\)-th roots \(\varpi ^{1/\ell ^{n}}\) of a uniformizing element \(\varpi \) of \(K\) and requiring \(\sigma (\varpi ^{1/\ell ^{n}}) = t_{ \ell ,n}(\sigma )\varpi ^{1/\ell ^{n}}\) for \(\sigma \in I\).
 
20
“Essentially known” means that the conjecture is known for varieties that are obtained from varieties over a global field of characteristic \(p\) whose completion at some place is \(K\). This is a result by Deligne himself.
 
21
In general, \(R\) is not a direct summand as the example \(R = \mathbb {Q}[X,Y]/(XY)\) and \(S\) its normalization shows: the map \(R \to S\) does not stay injective modulo \(X + Y\).
 
22
If this field has characteristic 0, then this is an easy trace map argument that shows that the conjecture holds in this case more generally if \(R\) is only assumed to be normal.
 
23
One then says that \(\operatorname{Spec}A \to \operatorname{Spec}R\) is a pro-étale morphism of schemes. But this notion of pro-étale is badly behaved (for instance it is not local on the target) and hence it is often better to work with the notion of being weakly étale ([56, Tag 092A]). Bhatt and Scholze show in [7] that both notions yields the same theory of sheaves and the same cohomology groups.
 
24
It remains the problem that cohomology with values in \(p\)-torsion sheaf or in \(\mathbb {Z}_{p}\), where \(p\) is a prime not invertible on the scheme. This is of principal nature, see the section on prismatic cohomology below.
 
25
A topological space \(S\) is called extremally connected if it is compact Hausdorff and the closure of every open set is open. Equivalently, \(S\) is a projective object in the category of compact Hausdorff spaces.
 
26
In fact, such an affine scheme \(Z = \operatorname{Spec}A\) is noetherian if and only if \(A\) is the product of a finite number of strictly henselian noetherian local rings. Then \(Z^{c}\) is a finite discrete space.
 
27
One has to be careful about the precise meaning of the inverse limit. It is not the inverse limit in the category of adic spaces. See [51, 6.4] or [52, 4.1] for a precise definition.
 
28
All coverings are always defined to be subject to a standard quasi-compactness assumption that ensures that one can refine coverings of quasi-compact spaces always by a finite covering of quasi-compact spaces. This condition also appears in the definition of the fpqc topology of schemes. For topologies whose coverings are defined to be surjective families of certain open morphisms (such as the étale or the fppf topology) this condition is automatic.
 
29
In fact, the diamond functor is also fully faithful on the category of seminormal rigid analytic spaces as defined in [32, 3.7]. This is useful since the pro-étale site of a rigid analytic space and of its semi-normalization are isomorphic by [32, 8.2.2].
 
30
If one views \(H^{n}(X,\mathbb {C})\) as cohomology in the constant sheaf, it is simply given by the map of complexes of sheaves of ℂ-vector spaces \(\mathbb {C}\to \varOmega ^{\bullet }_{X}\) which is a quasi-isomorphism by the Poincaré lemma for complex manifolds.
 
31
Let us remark on the Tate twists. If \(X\) is only defined over \(C\), then there is no Galois action and hence it is superfluous to include Tate twists. But very often, \(C\) is the completion of an algebraic closure \(\bar{K}\) of a discretely valued field \(K\) and \(X\) is already defined over \(K\). Then \(\varGamma _{K} = \operatorname{Gal}(\bar{K}/K)\) acts on \(H^{i+j}(X_{\operatorname{\'{e}t}},\mathbb {Z} _{p})\). Moreover, the action of \(\varGamma _{K}\) on \(\bar{K}\) extends by continuity to \(C\) making \(C\) into \(\varGamma _{K}\)-module, and the limit term of the spectral sequence is endowed with the tensor product action. Introducing the Tate twists on the initial terms then makes the whole spectral sequence \(\varGamma _{K}\)-equivariant. Moreover, the filtration on the limit term given by the spectral sequence does then split as \(\varGamma _{K}\)-modules over \(C\) ([46, 1.8]).
 
32
More generally, everything that is stated here still holds if \(K\) is an arbitrary complete discretely valued field extension of \(\mathbb {Q}_{p}\) with perfect residue field and if \({\mathscr{X}}\) is a smooth proper formal scheme over \(\operatorname{Spf}(O_{K})\) with \(X\) its rigid analytic generic fiber (same references). Moreover, some of the results stated below also hold if \({\mathscr{X}}\) is a proper, flat, and semistable scheme over \(O_{K}\) by work of Česnavičius and Koshikawa ([16]). Conjecturally, for every smooth proper variety \(X\) over \(K\) there exists such a model \({\mathscr{X}}\). In this semistable case one has to replace de Rham cohomology by logarithmic de Rham cohomology.
 
33
This is a very rich structure compared to the one on the integral \(\ell \)-adic cohomology \(H^{i}((X_{C})_{\operatorname{\'{e}t}},\mathbb {Z}_{\ell })\) (\(\ell \ne p\) a prime) that can be identified with \(H^{i}(({\mathscr{X}}_{\bar{k}})_{\operatorname{\'{e}t}},\mathbb {Z}_{\ell })\) and whose \(\varGamma _{K}\)-action factors through the Galois group of \(k\) as explained in Sect. 4.1. This is not true for \(H^{i}((X_{C})_{\operatorname{\'{e}t}},\mathbb {Z}_{p})\) on which usually the most interesting part of \(\varGamma _{K}\), the pro-\(p\)-part of the inertia group, acts non-trivially.
 
34
Here the Krull dimension of \(\operatorname{Spec}R\) is meant and we do not worry about its various pathologies, such as that \(\operatorname{Spec}R\) might be infinite dimensional.
 
35
Such prisms are called bounded in [9]. In other words we assume from now on that all prisms are bounded.
 
36
Everything below may be extended—although not verbatim—to not necessarily smooth \(p\)-adic formal schemes \(X\), see [9, 7.2].
 
37
This is only a technical assumption to avoid introducing more notation.
 
38
But note that both results are shown in their affine version in [9] without using the Hodge-Tate comparison and are used as ingredients to prove the Theorem 8.3.
 
39
This crystalline comparison theorem is proved in [9] before the Hodge-Tate comparison theorem and it is used as an ingredient of its proof.
 
40
To make “specializing” precise one should work with the complex https://static-content.springer.com/image/art%3A10.1365%2Fs13291-019-00211-y/MediaObjects/13291_2019_211_IEq1383_HTML.gif in the derived category of \(A\)-modules whose cohomology prismatic cohomology and consider derived base change.
 
41
Even simplicial algebra.
 
42
The class rings that are are flat over \(\mathbb {Z}_{p}\) or over some \(\mathbb {Z}/p^{n}\mathbb {Z}\) and that are quasi-syntomic (e.g., any noetherian \(p\)-complete locally complete intersection ring is quasi-syntomic).
 
43
This bijection has to preserve certain invariants. More precisely, it should match eigenvalues of Frobenius elements with Satake parameters.
 
44
Irreducible continuous, almost everywhere unramified and de Rham at places dividing \(p\).
 
45
\(L\)-algebraic, cuspidal, automorphic.
 
46
Compatible with twists, central characters, duals, \(L\)- and \(\epsilon \)-factors in pairs.
 
47
Supercuspidal, irreducible, smooth.
 
48
The Weil group is a variant of the Galois group that allows Frobenius elements in the Galois group of \(F\) (as explained in Sect. 4.1) to act arbitrarily on continuous representations over the complex numbers.
 
49
More precisely, a finite disjoint union of hermitian symmetric domains.
 
50
At least if \(K\) is sufficiently small. Otherwise one should work with Deligne-Mumford stacks if one does not want limit oneself to coarse moduli spaces.
 
51
Scholze conjectures in [48] that similar results should hold for arbitrary Shimura varieties. Shen generalized Scholze’s results to the large class of Shimura varieties of abelian type in [55].
 
52
In a suitable sense, see [52, 4.1] for a thorough discussion.
 
53
More precisely, it consists of those points whose rank-1-generizations specialize to a supersingular point of the special fiber.
 
54
Cuspidal automorphic representations \(\pi \) such that \(\pi _{\infty }\) is regular \(L\)-algebraic.
 
55
The (generalized) Ramanujan conjecture predicts that all local components of cuspidal automorphic representations are tempered.
 
56
But there is no good rigorous notion of dimension in this setting!
 
57
More precisely, a discrete Langlands parameter of \(G\).
 
58
The cocharacter is assumed to be minuscule in [41] but for some of the constructions explained below this hypothesis is not necessary.
 
59
Strictly speaking, this is only true if one formulates the entire theory for general not necessarily quasi-split reductive groups \(G\). If \(G\) is quasi-split, then \(J_{b}\) is only quasi-split if \(J_{b} = G\), which is usually not the case.
 
60
Or at least its connected component is assumed to be parahoric.
 
61
The category of compactly generated spaces is the “convenient category for homotopy theory” (often one also adds the separation axiom of being weakly Hausdorff). Recall that a topological space \(X\) is compactly generated if a subset \(A\) of \(X\) is closed if and only if for every continuous map \(f\colon S \to X\) with \(S\) compact Hausdorff (or, equivalently, profinite) \(f^{-1}(A)\) is closed. Every first countable space and every Hausdorff locally compact space is compactly generated.
 
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Metadata
Title
On the Work of Peter Scholze
Author
T. Wedhorn
Publication date
30-10-2019
Publisher
Springer Berlin Heidelberg
Published in
Jahresbericht der Deutschen Mathematiker-Vereinigung / Issue 4/2019
Print ISSN: 0012-0456
Electronic ISSN: 1869-7135
DOI
https://doi.org/10.1365/s13291-019-00211-y

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