Introduction to Rigid Analytic Geometry

Marc Masdeu

Abstract These are notes from a 90-minutes talk given in the study group at Warwick in November 2016, on an introduction to rigid analytic geometry. The main references used where [Sch98], [BCD+08, Chapter 1] and [Bos14].

1 Motivation

When studying algebraic varieties over a 푝-adic field (say over C푝), we would like to use analytic techniques. This is very fruitful when the base field is C, which allows many results to be transferred back and forth (using Serre’s GAGA) between the algebraic and analytic worlds. The main problem is that the (non-archimedean) on 푋 = C푝 is totally discon- nected. This makes the analysis difficult (for example, for any open 푈 ⊆ C푝, the space of locally constant functions on 푈 is of infinite dimension over C푝!). We will restrict the opens that we consider and, more importantly, we will also restrict types of “open coverings” that we allow. Let us recall how schemes are constructed: 1. For each 푅, we consider 푋 = Spec 푅 = {p prime ideals of 푅}, first as a set. We topologize 푋 = Spec 푅 by decreeing that a basis of closed sets is given by 푉 (퐼) = {p | 퐼 ⊆ p} ∼= Spec(푅/퐼). There is a convenient basis for this topology. Namely, for each 푓 ∈ 푅, we consider 푉 (푓) = Spec(푅/(푓)) ⊂ 푋 and 푋푓 = 푋 ∖ 푉 (푓) = {p | 푓 ̸∈ p}. (the sets 푋푓 are called “principal opens”).

2. Note that 푋 ∖ 푉 (퐼) = ∪푓∈퐼 푋푓 . 3. One remarks that 푅 can be thought of as the ring of functions on 푋 = Spec(푅): given 푔 ∈ 푅, there is a map ⋃︁ 푋 → 푘(p), p ↦→ image of 푔 in 푘(p). p∈푋

For example, if 푘(푥) = 푅푥/m푥 = C for all 푥 ∈ 푋, then this would be a function to C. . . In a similar fashion, for each 푓 ∈ 푅 we obtain 푋푓 = Spec(푅푓 ), and so 푅푓 can be thought of as the ring of functions of a principal open.

1 4. Our goal is to define, for each open set 푈 ⊂ 푋, a ring 풪푋 (푈) of “functions on 푈”. So far, we have defined 풪푋 (푋) = 푅 and 풪푋 (푋푓 ) = 푅푓 (localizations). However, although 푋푓푔 = 푋푓 ∩ 푋푔 and so the set of principal opens is closed under finite intersection, it is not closed under unions, so we need to extend the definition of 풪푋 a bit more. This is quite easy to do, provided that we have what is called the axiom: given a basic

open 푋푓 and a covering by it by basic opens 푋푓 = ∪푖∈퐼 푋푓푖 , then the following diagram (of sets, or of abelian groups) must be exact: ∏︁ ∏︁ 풪푋 (푋푓 ) → 풪푋 (푋푓푖 ) → 풪푋 (푋푓푖 ∩ 푋푓푗 ). 푖∈퐼 푖,푗 One checks that this is in fact true. The crucial fact that one needs to use is that 푋푓 = Spec(푅푓 ) is quasi-compact. This allows one to extract a finite subset 퐽 ⊂ 퐼 and work with only those opens, to glue the sections.

5. Finally, one defines a as a 푋 and a sheaf of rings 풪푋 on 푋 ⋃︀ such that 푋 = 푖∈퐼 푈푖 with (푈푖, 풪푋 |푈푖 ) = Spec(퐴푖). We will try to mimic this construction in the non-archimedean world (so the rings to be considered will be rings of convergent power series, instead of the usual polynomial rings used in schemes).

2 Affinoid algebras

Fix a non-archimedean field 푘 with valuation ring 푅 and residue field 푘˜ = 푅/m. The Tate algebra over 푘 in 푛 variables is {︂ }︂ ∑︁ 퐽 ∑︁ 푇푛 = 푇푛(푘) = 푘⟨푋1, . . . , 푋푛⟩ = 푎퐽 푋 | |푎퐽 | −→ 0 , ‖퐽‖ = 푗푖. ‖퐽‖→∞

The Gauss norm on 푇푛 is:

∑︁ 퐽 ‖ 푎퐽 푋 ‖ = max |푎퐽 |. 퐽 It enjoys the following basic properties: ∙‖ 푓 + 푔‖ ≤ max(‖푓‖, ‖푔‖).

∙‖ 푐푓‖ = |푐|‖푓‖.

∙‖ 푓푔‖ = ‖푓‖‖푔‖.

2 ∙ 푇푛 is complete for the induced metric.

∙‖ 푓‖ = sup푥∈퐵(푘) |푓(푥)| = max푥∈퐵(푘) |푓(푥)|. (Here 퐵(푘) = {푥 ∈ 푘 | |푥푖| ≤ 1}).

The first properties make (푇푛, ‖·‖) into a 푘-. Note that the function 푓(푥) = 푥 − 푥푝 has the property that ‖푓‖ = 1, but sup |푓(푥)| = 1/푝. So for the maximimum 푥∈Z푝 principle to hold it is important to allow evaluation at elements in the algebraic closure. The Tate algebra 푇푛 is a “free object” in the sense that

{continuous 푘-algebra maps 푇푛 → 푘} ↔ 퐵푛(푘)

The next step is to prove a form of Weierstrass Preparation in this setting, from which one can show:

Proposition 1. ∙ 푇푛 is a Noetherian, regular and UFD.

∙ For every maximal m, the localization (푇푛)m has dimension 푛 and its residue field 푇푛/m is a finite extension of 푘.

∙ 푇푛 is Jacobson: every prime is the intersection of maximal ideals.

∙ Every ideal in 푇푛 is closed with respect to the Gauss norm. As a consequence, we get ‖푓‖ = sup |푓(푥)|, 푥∈Max(푇푛) which allows us to avoid speaking of 푘, and also shows that ‖푓‖ is an intrinsic norm, inde- pendent of the choice of “coordinates” 푥1, . . . , 푥푛.

Affinoid algebras

An affinoid algebra isa 푘-algebra 퐴 isomorphic to 푇푛/퐼 for some ideal 퐼 ⊂ 푇푛. Write 푀(퐴) = Max(퐴). Choosing one such isomorphism we get an embedding 푀(퐴) ˓→ 푀(푇푛), with image {푥 ∈ 푀(푇푛) | 푓(푥) = 0 for all 푓 ∈ 퐼}.

Note that things start getting a bit more complicated, though: firstly, 푀(푇푛) (and also 푀(퐴)) have a lot more points than those coming from 푅푛. Remark. Note that 퐴⟨푥⟩/(푥−푎) is not the same as 퐴 in general! In fact 푀(퐴⟨푎⟩) is identified with the subset of 푀(퐴) consisting of elements 푥 such that |푎(푥)| ≤ 1.

3 However, some sanity is preserved. Every 푘-affinoid algebra 퐴 is noetherian and Jacobson of finite Krull dimension, and 퐴/m = 푘(m) is a finite extension of 푘 (for each m ∈ 푀(퐴)). Also, a function 푎 ∈ 퐴 is nilpotent if and only if 푎(푥) = 0 for all 푥 ∈ 푀(퐴) (by Jacobson property), and 푎 ∈ 퐴× if and only if 푎(푥) ̸= 0 for all 푥 ∈ 푀(퐴). So we can think of elements of 퐴 as functions on 푀(퐴) (valued in fields that vary at each points). Affinoid algebras admit a 푘-Banach algebra structure (using the residue norm) that is ∼ defined using a presentation 퐴 = 푇푛/퐼. In fact, this definition depends on the presentation, but the topology induced from it does not. So concepts like boundedness still make sense. There is a maximum modulus principle, which can be interpreted as saying that 푀(퐴) “wants to be compact”: for all 푓 ∈ 퐴, we have

‖푓‖sup := sup |푓(푥)| = max |푓(푥)| < ∞. 푥∈푀(퐴) 푥∈푀(퐴)

If 퐴 is reduced, the above gives a 푘-Banach structure on 퐴 which is intrinsic, and recovers the Gauss norm when 퐴 = 푇푛. The only catch is that it is not multiplicative, in this case (only power-multiplicative).

Lemma 2. If 푎 ∈ 퐴, then 푎 is power-bounded if and only if |푎(푥)| ≤ 1 for all 푥 ∈ 푀(퐴).

The canonical topology We want to put a Hausdorff topology on 푀(퐴). This can be done as follows:

∙ Let 퐴(푘) denote the set of 푘-algebra maps 퐴 → 푘 with image contained in a finite extension of 푘. The Galois group 퐺푘 = Gal(푘/푘) acts on it by composition. ∙ For each 푥 ∈ 푀(퐴), choosing an embedding 푖: 푘(푥) ˓→ 푘 gives an element in 퐴(푘). Changing the embedding gives an element which is in the same 퐺푘-orbit. Hence this gives a well-defined map 푀(퐴) → 퐴(푘)/퐺푘, which turns out to be a bijection. All this is also functorial in 퐴.

∙ Put on 퐴(푘) a topology by taking as basis of open sets the sets of the form

{푥 ∈ 퐴(푘) | |푓푖(푥)| ≥ 휖푖, |푔푗(푥)| ≤ 휂푗, ∀푖, 푗}.

∙ Give 푀(퐴) the quotient topology.

One checks that the resulting topology is Hausdorff, totally disconnected, and functorial in 퐴. This is called the canonical topology.

4 Laurent, Weierstrass and rational domains If 푋 = 푀(퐴) is an affinoid domain (that is, 퐴 is an affinoid algebra), then a Laurent domain is a set of the form −1 푋⟨푓, 푔 ⟩ = {푥 ∈ 푋 | |푓푖(푥)| ≤ 1, |푔푗(푥)| ≥ 1}. This in fact is 푀(퐴⟨푓, 푔−1⟩), where 퐴⟨푋, 푌 ⟩ 퐴⟨푓, 푔−1⟩ = . (푋1 − 푓1, . . . , 푔1푌1 − 1,...)

These have a universal property, in the same way that the Tate algebras 푇푛 have as well: for any map of 푘-Banach algebras 휑: 퐴 → 퐵, the diagram

퐴 / 퐴⟨푓, 푔−1⟩ 휑

 $ 퐵 can be filled in in at most one way, and such an arrow exists if and onlyif 휑(푓푖) are power- × −1 bounded for all 푖, and 휑(푔푗) ∈ 퐵 and 휑(푔푗) are power-bounded for all 푗. The set 푋 = 푀(퐴) is called a Weierstrass domain in the special case that there are no 푔푗’s. All of these play the role of “opens” in our theory, as we will see. Next, let 푓1, . . . , 푓푛 and 푔 be elements of 퐴 with no common zero. Then we define a rational domain as the set

푋⟨푓1/푔, . . . , 푓푛/푔⟩ = {푥 ∈ 푋 | |푓푖(푥)| ≤ |푔(푥)|, ∀푖}, which corresponds to the affinoid algebra

퐴⟨푋1, . . . 푋푛⟩ 퐴⟨푓1/푔, . . . , 푓푛/푔⟩ = . (푔푋1 − 푓1, . . . 푔푋푛 − 푓푛)

Affinoid subdomains The previous were just examples of affinoid subdomains. They will be important because rational domains will play the role of principal opens, but we want define the opens of interest independently. Definition 3. Let 퐴 be an affinoid algebra. A subset 푈 ⊂ 푀(퐴) is an affinoid subdomain if there is a map 퐴 → 퐴푈 of 푘-affinoids (for some 퐴푈 ) such that 푀(퐴푈 ) → 푀(퐴) has image in 푈 and is universal for this condition. Precisely, if 휑: 퐴 → 퐵 is any map of 푘-affinoids such that 푀(퐵) → 푀(퐴) has image in 푈, then there is a unique map 퐴푈 → 퐵 making the corresponding diagram commutative.

5 The affinoid algebra 퐴푈 (unique by Yoneda) is called the coordinate ring of 푈. Theorem 4 (Gerritzen–Grauert). Every affinoid subdomain 푈 ⊆ 푀(퐴) is a finite union of rational domains.

Note that the converse is not true in general. It is hard to determine when such a union is affinoid. . It is easy to show that the special cases of affinoid subdomains of the previous subsection are all open for the canonical topology, so the previous Theorem gives in particular that all affinoid subdomains are open. In fact, Weierstrass domains form a basis for the canonical topology. Remark. During the talk we discussed whether a singleton {푥} ⊂ 푀(퐴) as an affinoid sub- domain. It turns out that it is not: if one considers the map 퐴 → 퐴/m푥 (the one we had in mind during the discussion), then it fails to satisfy the universal property for the affinoid 2 2 푘-algebra 퐵 = 퐴/m푥: the map 퐴 → 퐴/m푥 does not factor in general through 퐴/m푥.

3 A 퐺-topology

Let 퐴 be a 푘-affinoid algebra, and let 푋 = 푀(퐴). We want to restrict the set of opens we consider. One can take as the admissible opens the affinoid subdomains, and allow only those coverings which consist of finitely many affinoid subdomains. This is what is called the weak topology. In order to globalize it is better to allow more opens and more coverings. So we “slightly refine” this topology as follows.

Definition 5. A subset 푈 ⊆ 푋 is an admissible open if it can be covered by affinoid sub- domains 푈푖 ⊆ 푋 such that for every map 휑: 퐴 → 퐵 satisfying 푀(휑)(푀(퐵)) ⊆ 푈, the open −1 covering {푀(휑) (푈푖)} of 푀(퐵) has a finite subcovering. Next we restrict the possible coverings that we can consider.

Definition 6. Let {푉푗} be a collection of admissible opens. This is an admissible covering of the union 푉 = ∪푉푗 if for every map 휑: 퐴 → 퐵 with 푀(휑)(푀(퐵)) ⊆ 푉 , the covering −1 {푀(휑) (푉푗)} of 푀(퐵) has a finite refinement consisting of affinoid subdomains.

Remark. ∙ It follows that 푉 above will be admissible! Just cover each of the 푉푗 by affinoid subdomains.

∙ Also, if 푈 is admissible then the covering appearing in its definition is automatically an admissible covering, too.

6 ∙ Finally, finite unions of affinoid subdomains are admissible.

Let us see how all this work starts to give some results. Let 푇1 = Q푝⟨푡⟩. We have the Laurent domain (hence admissible) 푉 = {|푡| = 1} and also the open unit ball 푈 = {|푡| < 1}. It turns out that 푈 is also admissible. In fact,

−1/푛 푈 = ∪푛≥1푈푛, 푈푛 = {|푡| ≤ 푝 }.

Note that 푈푛 are Weierstrass domains, and the maximum modulus principle guarantees that this makes 푈 admissible. However, the covering 푋 = 푈 ∪ 푉 is not admissible. Applying the condition to 휑 being the identity map would yield a refinement of the covering {푈, 푉 }. The maximum modulus principle would give that 푋 can be covered by 푉 and 푈푛 for some sufficiently large 푛, and this is clearly a contradiction because we can take arbitrarily ramified extensionf of Q푝. We define the Tate topology as the 퐺-topology that has as objects the admissible opens and as coverings the admissible open coverings. So far we have bought ourselves the chance that the closed unit ball may be connected. . .

Example (A non-admissible open). Let 퐴 = 푘⟨푥, 푦⟩, and pick 푐 ∈ 푘× with 푟 = |푐| satisfying 0 < 푟 < 1. Define

⋃︁ 푖 1/푖 푈 = 푉 ∪ 푈푖, 푉 = {|푦| = 1}, 푈푖 = {|푥| ≤ 푟 , |푦| ≤ 푟 }. 푖≥1

Note that 푈 ⊃ {푥 = 0}. If 푈 were admissible, it would contain a tube1 of the form {|푥| ≤ 푟푖0 }. In this case, let 푏 ∈ 푘¯ with 푟푖0 < |푏| < 1.

Then the point (푐푖0 , 푏) belongs to the tube but is not in 푈, which is a contradiction.

In order to have a sheaf theory for the Tate topology we need to be able to glue uniquely sections on different opens. This was proved by Tate and is the crucial result that getsus going.

Theorem 7 (Tate’s Acyclicity theorem). Let 퐴 be a 푘-affinoid algebra. The assignment 푈 ↦→ 퐴푈 extends to a sheaf 풪퐴 for the Tate topology on 푀(퐴). The proof of this theorem proceeds by formally reducing to the special case of the covering 푀(퐴) = 푀(퐴⟨푓⟩) ∪ 푀(퐴⟨1/푓⟩), for 푓 ∈ 퐴. He then did a direct computation.

1We have not proved this fact. . .

7 4 Examples

4.1 Affine space

While the unit polydisk is here the “trivial example” (just take B푛(1) = Sp(푇푛(푘))), one may be a little upset by the fact that the “model” space is not the affine space, but just a polydisk. That is a fact of life, but at least it is satisying to know that affine space can also be constructed as a rigid analytic space. The idea is to realize that A푛 is the union of polydisks of increasing radii. That is, for each 푟 ≥ 1, we may consider

푛 B푛(푟) = {푥 ∈ A | |푥| ≤ 푟}, which looks like one would be able to construct as easily as B푛(1). Taking the union of these 푛 balls gives A = ∪푟B푛(푟). The glueing is done as follows (restrict to 푛 = 1 for ease of notation): pick 푐 ∈ 푘 with 0 < |푐| < 1, and let 퐷푗 = B1(1) with coordinate 푥푗. Map 퐷푗 → 퐷푗+1 via 푥푗+1 ↦→ 푐푥푗. Therefore 퐷푗 is identified with an affinoid subdomain of 퐷푗+1, namely with {푥푗+1 ≤ |푐|}. 푗 In particular, the analytic function 푥푗/푐 are all compatible when moving 푗. This yields a 푗 1,an global section 푥 in the glueing, satisfyig 푥|퐷푗 = 푥푗/푐 . We have just constructed A푘 , the −푗 rigid affine line over 푘. The locus {|푥| ≤ |푐| } is identified with the open subspace 퐷푗. 푛,an One constructs in the same way A푘 . This deserves to be called affine 푛-space because it satisfies the following universal property: if 푋 is any rigid space, the coordinates 푥1, . . . , 푥푛 give a bijection

푛,an 푛 ♯ ♯ Hom(푋, A푘 ) → 풪푋 (푋) , 푓 ↦→ (푓 (푥1), . . . , 푓 (푥푛)).

4.2 Projective space We could proceed to construct projective space by glueing copies of affine space in the usual way. However, it is more direct and useful to directly glue polydisks. For example, to 1,an construct P푘 we may take two unit balls and glue them along the “boundary” as well. Concretely, take balls 퐵1 and 퐵2 with coordinates 푥1 and 푥2. We may glue them along the −1 ∼ −1 −1 isomorphism 퐵1⟨푥1 ⟩ = 퐵2⟨푥2 ⟩, as usual (i.e. identifying 푥1 = 푥2 ).

4.3 Analytic tori

Let 푞 ∈ 푘 with 0 < |푞| < 1. Consider the affinoid subdomains of 푋 = B1 = Sp(푘⟨푡⟩):

−1 −1 −1/2 −1 1/2 −1 −1 푈 = 푋⟨푞 푡, 푞푡 ⟩, 푋1 = 푋⟨|푞| 푡, 푞푡 ⟩, 푋2 = 푋⟨|푞| 푡 ⟩, 푉 = 푋⟨푡 ⟩.

8 Take now 푋1 ∪ 푋2, and identify 푈 ⊂ 푋1 with 푉 ⊂ 푋2 via the isomorphism 휙: 푈 → 푉 corresponding to 휙* : 푘⟨푡, 푡−1⟩ → 푘⟨푞−1푡, 푞푡−1⟩, 푡 ↦→ 푞−1푡. Z The resulting rigid analytic space, written G푚,푘/푞 , is the one-dimensional rigid analytic torus defined by 푞.

Theorem 8 (Tate). Let 퐸/푘 be an with 푗-invariant 푗퐸 satisfying |푗퐸| > 1. an ∼ Z Then there is a Galois-equivariant isomorphism of rigid analytic spaces 퐸 = G푚,푘/푞퐸, −1 where 푞퐸 ∈ 푘 satisfies 푗(푞퐸) = 푞퐸 + 744 + 196884푞퐸 + ··· = 푗퐸. Conversely, if 푞 ∈ 푘 satisfies 0 < |푞| < 1, there is an isomorphism of rigid analytic varieties Z an 휑: G푚,푘/푞 → 퐸푞 , an where 퐸푞 is the analyfication of the elliptic curve 5푠 (푞) + 7푠 (푞) 퐸 : 푦2 + 푥푦 = 푥3 + 푎 (푞)푥 + 푎 (푞), 푎 (푞) = −5푠 (푞), 푎 (푞) = − 3 5 , 푞 4 6 4 3 6 12 with ∑︁ 푛푘푞푛 푠 (푞) = ∈ [[푞]]. 푘 1 − 푞푛 Z 푛≥1

5 GAG(R)A

It is natural to wonder about the relationship that this newly introduced theory has with the theory of schemes over 푘. Recall that in the more classical setting of 푘 = C, Serre proved theorems that related proper algebraic varieties over C with complex-analytic varieties, in a way that the sheaf theory matched. The same happens in our setting. There is a unique morphism (of locally ringed 퐺-topologized spaces equipped with a 푘-algebra structure sheaf) 푛,an 푛 푖 푛 : → , A푘 A푘 A푘 which is compatible with the standard 푛-tuple of global functions on each space. The respec- tive universal properties of these spaces give that this map is final amongst all rigid analytic 푛 푛 spaces with maps to A푘 . This is called an analytification of A푘 . An argument using coherent ideal sheaves allows to extend this result to all affine algebraic 푘-schemes of finite type, and then to all 푘-schemes locally of finite type by a glueing argument. We obtain a functor 푎푛 Sch푘 → RigAn푘, 푋 ↦→ (푖푋 : 푋 → 푋) from schemes to rigid analytic space with a morphism to schemes, in a way that the coherent sheaf theory (and the corresponding cohomology theory) is “the same”.

9 6 Raynaud’s formal models

Sometimes one needs to deal with more algebraic questions, for which rigid analytic geometry is ill-suited. Consider a topologically finitely presented (tfp) 푅-algebra 풜:

풜 = 푅{푋1, . . . , 푋푛}/ℐ, where 푅{푋} denotes restricted power series (coefficients tending to 0), and ℐ is a finitely- generated ideal. The tfp 푅-algebra 풜 is called admissible if it is 푅-flat. Remark. 푅{푋} is the 휋-adic completion of 푅[푋], with 휋 ∈ 푅 satisfying 0 < |휋| < 1. If 푘 is discretely valued this is the same as the m-adic completion. But if 푘 is algebraically closed then these two are very different! The m-adic completion would give 푘˜[푋].

Note that 푘 ⊗푅 푅{푋} = 푇푛(푘), so one may think of tfp 푅-algebras as “integral” versions of the affinoid algebras. Lemma 9 (Existence of integral models). Let 퐴 be a 푘-affinoid algebra. Then there exists an admissible 푅-algebra 풜 such that 퐴 = 푘 ⊗푅 풜. One can constructs a theory of tfp formal schemes X over 푅, by glueing Spf of the above (with an appropriate structure sheaf). As before, admissible tfp formal schemes mean that the coordinate ring of any affine open is 푅-flat. Essentially by extending scalars to 푘, one defines a functor X ; X푘, known as “Raynaud’s generic fiber” functor. Theorem 10 (Raynaud). There is an equivalence of categories between {formal flat 푅-schemes tfp} /blow-up ←→ {quasi-compact, quasi-separated rigid 푘-analytic varieties} .

References

[BCD+08] Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran S. Kedlaya, and Jeremy Teitel- baum. 푝-adic geometry, volume 45 of University Lecture Series. 2008. [Bos14] Siegfried Bosch. Lectures on formal and rigid geometry, volume 2105 of Lecture Notes in Mathematics. Springer, Cham, 2014. [Pap05] Mihran Papikian. Rigid-analytic geometry and the uniformization of abelian varieties. In Snowbird lectures in , volume 388 of Contemp. Math., pages 145–160. Amer. Math. Soc., Providence, RI, 2005. [Sch98] Peter Schneider. Basic notions of rigid analytic geometry. LMS lecture note series, pages 369–378, 1998.

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