This is the first in a series of 5 posts whose goal is to briefly introduce rigid geometry with a focus on providing a big picture between the interactions between different perspectives of rigid geometry.

Motivation

We shall not go into the motivation for rigid geometry in any serious depth here since this has been done ad infinitum in other sources ([Con1], [Nic], [FK1],…) . But, we would be remiss to not say anything at all.

One of the most fruitful interactions in the semi-classical theory of algebraic geometry is that between the theory of schemes and the theory of complex analytic varieties. Namely, there is an analytification functor

$\left\{\begin{matrix}\text{Finite type schemes}\\ \text{over }\mathbb{C}\end{matrix}\right\}\to \left\{\begin{matrix}\text{Complex analytic}\\ \text{spaces}\end{matrix}\right\}$

which one can read about, for example, in the book [Nee]. Intuitively, this analytification functor takes a $\mathbb{C}$ -scheme $X$ to $X(\mathbb{C})$ which is endowed with the natural structure of a $\mathbb{C}$ -manifold (it’s really only a manifold if $X$ is smooth) by taking the algebraic charts and interpretting them as holomorphic charts.

It turns out that this functor preserves many of the properties of $\mathbb{C}$ -schemes algebraic geometers are interested in (cf. [SGA1, Exposé XII)). This statement is no better exemplified than in the GAGA results of Serre et al. (e.g. as in [Ser]) as well as the relationship betwen the étale topology of $X$ and the analytic topology of $X^\mathrm{an}$ exhibited by Grothendieck, Artin, et al. (e.g. see [SGA1, Exposé XII Théorème 5.1] and [SGA4, Exposé XI Théorème 4.4]).

This connection has immense benefits for both the theory of schemes and the theory of complex analytic spaces. For example, one has access to a much wider class of objects in the world of complex analytic spaces that don’t exist in the purely algebro-geometric world. As a simple example of this, in the complex analytic world one can make arguments using discs which are convenient for topological arguments since they’re contractible, whereas such arguments aren’t feasible in the purely scheme theoretic situation. This leads to situations where many results about varieties over characteristic $0$ fields are reduced to statements in the complex analytic setting by the so-called Lefschetz principle which are then proved by complex analytic means. For example, for a long period of time the only known proofs of the degeneration of the Hodge spectral sequence (e.g. see this post) and of the Kodaira-Nakano-Akizuki vanishing theorem (cf. loc. cit.) were only known by an application of the Lefschetz principle.

Now, while the the reduction to $\mathbb{C}$ is useful for proving purely geometric results, the passage from something like $\mathbb{Q}_p$ to $\mathbb{C}$ via the Lefschetz principle completely forgets arithmetic. Namely, if $X$ is a variety over $\mathbb{Q}_p$ then the variety $X_{\overline{\mathbb{Q}_p}}$ has an action of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ which is of great interest to number theorists. Such information is completely lost in the passage from $\mathbb{Q}_p$ to $\mathbb{C}$ .

To remedy this one might note that $\mathbb{Q}_p$ , like $\mathbb{C}$ , is an ‘analytic field’—it has a natural metric for which it is a complete topological field. So, one might ask whether or not there is some sort of $p$ -adic analytification functor which takes varieties over $\mathbb{Q}_p$ and outputs ‘ $p$ -adic analytic manifolds’. The hope is that such $p$ -adic analytic manifolds would have the benefits of complex analytic manifolds (e.g. access to disks) while retaining the arithmetic information contained in the original variety.

The goal of rigid analytic geometry is, essentially, to create such a theory of $p$ -adic analytic manifolds. As it turns out such a theory is fraught with technical difficulties from the start. Most prominently is the fact that the naive $p$ -adic disk is totally disconnected which makes naive attempts at defining a theory unwieldy, bordering on usless. Thus, the actual structures involved in the theory of rigid analytic spaces are significantly more nuanced than one first might envisage—they are certainly not just ‘spaces which locally look like open subspaces of $\mathbb{Q}_p^n$ ‘ or the like (although there are theories of this form which are useful in contexts other than geometry — e.g. see [Sch]).

In fact, there are (more than) six theories of rigid analytic geometry, and we hope to touch on each of them in these posts. These are Tate’s theory of rigid analytic varieties, Raynaud’s theory of birational geometry of formal schemes, Berkovich’s theory of Berkovich analytic spaces, and Huber’s theory of adic spaces, and Fujiwara–Kato’s theory, and Scholze’s theory of diamonds (although this last one has a slightly different character than the others). In very broad strokes one can think roughly that

Tate’s theory is like the classical theory of varieties over an algebraically closed field, and roughly has the same advantages and pitfalls.
Huber’s theory is like the theory of schemes, and roughly also has the same advantages and pitfalls.
Berkovich’s theory is a middle ground between Tate’s theory and Huber’s theory which has some of the same pitfalls that Tate’s theory has, but with the advantage of being topologically well-behaved (in large generality Berkovich spaces are Hausdorff and arc connected!).
Raynaud’s theory tries to make precise the idea that rigid analytic varieties over $\mathbb{Q}_p$ should be thought of as the ‘generic fibers’ of formal schemes over $\mathbb{Z}_p$ . This has the advantage of making available the full theory of formal geometry (which has historically been well-developed) but has the disadvantage of being somewhat difficult to talk about simple objects—many non-isomorphic formal schemes have the same ‘generic fiber’ and so rigid analytic varieties in this perspective are something like equivalence classes of formal schemes.
Fujiwara–Kato’s theory chooses to take Raynaud’s theory to the next level by constructing something akin to the ‘Zariski–Riemann space’ of a formal scheme. Roughly this is something like an inverse limit over all formal schemes which have isomorphic generic fiber.
In the p-adic situation, Scholze’s theory of diamonds chooses to view rigid spaces in terms of their interaction with extremely degenerate, but extremely flexible spaces: the perfectoid spaces. This has the advantage of allowing one to reduce many proofs to arguments with a point-set topology theoretic flavor. The disadvantage (besides the inherent complicatedness of the objects involved) is that the theory of diamonds can only remember ‘topological information’ (e.g. the etale site).

All of these theories are relatively different in their presentation even though they generally capture, at their heart, the same theory (except for the theory of diamonds as already mentioned). In fact, one of the many strengths of rigid analytic geometry is that one can move between these various theories to use their different strengths. For example, there might be some powerful argument utilizing the well-behaved topology of Berkovich spaces that is completely invisible (or certainly much less obvious) from the perspective Huber.

The point of these posts

This series of posts is meant to introduce the reader to rigid geometry in terms of these perspectives: the classical Tate perspective on rigid spaces, the perspective of Raynaud and Fujiwara–Kato in terms of rigid fibers of formal schemes, the perspective of Berkovich spaces, Huber’s perspective in terms of adic spaces, and Scholze’s theory of diamonds.

These posts are mainly meant to be high-level introductions, focusing on giving some major definitions, major results, quirky examples and observations, but no real proofs. In particular, we will (of course) not go into any real detail here. For this, I highly suggest the following sets of notes/books:

For Tate perspective:

Tian’s notes [Tia]. These do a really great job of summarizing the key points of the theory and, in particular, strikes a great balance between terseness and rigor.
Bosch’s book [Bos]. This is the book Tian’s mostly drew from. It is also wonderful, if a bit longer than Tian’s notes. If one wants more detail than what is in [Tian] I suggest looking here.
The book [FvdP] by Fresnel and Van de Put is a pretty commonly cited place to look for this theory. It is a bit hard to read in places, but introduces most of the general theory in a good amount of rigor, tries to give some motivational examples first (by considering subspaces of $\mathbb{P}^{1,\mathrm{an}}_k$ ), and gets into more advanced topics (e.g. rigid and étale cohomology).
The book [BGR] by Bosch, Guntzer, and Remmert. This is a bit like an EGA for the basic theory of rigid spaces. It’s a great place to look for rigorous statements/proofs of technical results.
The first chapter of [Ayo] is a great place to get a high-level, rapid, but mostly complete introduction to the technical setup of rigid geometry from the Tate perspective.

For Raynaud’s perspective:

I again suggest [Tia], for the same reasons.
I again suggest [Bos], for the same reasons.
The book [FK] is essentially a reimagining/strengthening of the classical perspective of Raynaud. This book is only the first book in a series of books. It is written very, very much in an EGA format.

For Berkovich’s perspective:

I suggest the notes [Jon] as a good place to start.
Of course, the foundational texts [Ber] and [Ber1] are also a great place to look.
Part I and Part II of [DT] is also helpful.

For Huber’s perspective:

I suggest Huber’s papers [Hub1] and [Hub2] as well as Huber’s book [Hub] for much of the rigorous foundations. This is where I first learned most of this material, and remains the only place for many fundamental results (especially considering things about étale cohomology).
The book [SW] has a good introduction to adic spaces in Lecture 2 through Lecture 5.
The notes [Mor] contain many of the foundational results for adic spaces.
The series of lectures notes from [Con+] also contain much foundational material.

For Fujiwara–Kato’s perspective

The only real canonical reference is the first book on this topic by the authors. Namely, [FK].
One can also see the paper [FK1] which is a good conceptual introduction to the ideas which prompt the [FK] theory, and some of its advantages.

For Scholze’s theory of diamonds

The best introduction to these ideas is provided in [SW], where they began to take shape. This lacks many of the deep foundational results though.
These deep foundational results are then instead supplied in [Scho].
A nice overview of the theory is also provided in the first several lectures of Scholze’s course on Fargues–Scholze (see here).

General overview

One can look at the notes [Con1] of Conrad from the 2007 Arizona Winter School. They do a great job of introducing all the key players, but with a special emphasis on the Tate perspective.
One can also look at the notes [Nic] which are similar in scope to [Con1] but have a different set of biases for what to emphasize.
One can also see the text [FK1] which does a good job of discussing a broad swath of the rigid analytic theory.

The Tate perspective

As mentioned above, Tate’s theory of rigid analytic geometry is very similar in spirit to the classical theory of varieties over an algebraically closed fields $k$ —namely spaces locally isomorphic to closed subsets of $k^n$ . That said, there is an added twist in the Tate setting since the ‘spaces’ we’ll be considering aren’t actually topological spaces!

For the entirety of this post let us fix $k$ to be a non-archimedean complete (but not necessarily locally compact) field with respect to an absolute value $|\cdot |$ . Examples of such fields are

The field $\mathbb{Q}_p$ of $p$ -adic numbers with the usual $p$ -adic absolute value $|\cdot |_p$ (which we will just usually abbreviate to $|\cdot |$ ), or finite extensions thereof (with the unique extension of $|\cdot|_p$ ).
The field $\breve{\mathbb{Q}}_p$ which is the completion of the maximal unramified extension $\mathbb{Q}_p^\mathrm{ur}$ of $\mathbb{Q}_p$ (with respect to the unique extension of $|\cdot |_p$ ).
The field $\mathbb{C}_p$ which is the completion of $\overline{\mathbb{Q}_p}$ (with respect to the unique extension of $|\cdot|_p$ ).
The field $\mathbb{F}_p((t))$ with the usual $t$ -adic absolute value , or a finite extension thereof (with the unique extension of the $t$ -adic absolute value).
The field $\mathbb{C}((t))$ with the $t$ -adic absolute value.

Let us also fix an element $\pi$ of $k$ satisfying $0<|\pi|<1$ (such an element is called a pseudo-uniformizer since it plays the role for such $k$ that a uniformizer does for discretely valued fields.

The general setup

Prologue

As mentioned above, Tate’s perspective is the beginning of rigid geometry which, in essence, tries to capture an ‘algebraic theory of $p$ -adic manifolds’. By an ‘algebraic theory’ I mean that we want to consider our building blocks to come from ring theory of analytic functions, and our opens to be somewhat algebraic in nature. The big difference between the algebro-geometric setup and the rigid analytic setup is that in the latter instead of just considering opens to essentially be those generated by non-vanishing loci of analytic functions

$\{x:f(x)\ne 0\}$

we also want to consider opens given by inequalities involving analytic functions

$\{x:|f(x)|\leqslant 1\}$

(which, by the magic of non-archimedean theory, are open in the topology given by $|\cdot|$ ) and, of course, combinations of these two types of opens (which one might call something like semi-algebraic open subsets).

Remark 1: Note that since this will be an ‘algebraic theory’ of $p$ -adic manifolds some sort of surprising things will happen. For example, the closed unit disk over $\mathbb{C}_p$ , denoted $\mathbb{B}_{\mathbb{C}_p}$ , will be both connected and quasi-compact. These are necessary steps to make a good algebraic theory work. For example, we would imagine that the global ring of functions on $\mathbb{B}_{\mathbb{C}_p}$ is some ring of analytic functions (convergent power series) and, in particular, an integral domain. So, if $\mathbb{B}_{\mathbb{C}_p}$ is to have a sheaf $\mathcal{O}_{\mathbb{B}_{\mathbb{C}_p}}$ of analytic functions, the fact that $\mathcal{O}_{\mathbb{B}_{\mathbb{C}_p}}(\mathbb{B}_{\mathbb{C}_p})$ is an integral domain actually forces connectedness (e.g. see the argument in [Stacks, Tag00EC]). This is in stark contrast to the naive $p$ -adic manifold

$\mathbb{B}_{\mathbb{C}_p}(\mathbb{C}_p)=\mathcal{O}_{\mathbb{C}_p}=\{x\in\mathbb{C}_p:|x|\leqslant 1\}$

which, with its native $p$ -adic topology, is neither connected nor quasi-compact. $\blacklozenge$

Now, just as one can imagine building the theory of varieties in three steps:

(Step 1) Define affine space $\mathbb{A}^n_k$ .
(Step 2) Define affine varieties as closed subsets $V(I)\subseteq \mathbb{A}^n_k$ .
(Step 3) Define general varieties as gluings (along affine opens) of affine varieties.

one can also see Tate’s theory of rigid analytic varieties as following the same three step procedure with some differences. We will explain this progression, roughly, below.

Step 1

The first major difference is that instead of our building blocks being affine space, which intrinsically doesn’t capture our desired notion of inequalities, we instead use closed polydisks $\mathbb{B}^n_k$ which one imagines as

$\mathbb{B}^n_k:=\{(x_1,\ldots,x_n):|x_i|\leqslant 1\}$

(we will shorten $\mathbb{B}_k^1$ to $\mathbb{B}_k$ ) as our building blocks in Step 1.

Of course, the naive guess for what the closed unit polydisk $\mathbb{B}_k^n$ should be is that it’s, roughly, the set

$\{(x_1,\ldots,x_n)\in k^n:|x_i|\leqslant 1\}$

Of course, if $k$ is not algebraically closed this has the exact same pitfalls as if one considers $k$ to be the ‘affine line’ over $k$ . For example, the closed subspace $V(x^2-\alpha)$ , where $\sqrt{\alpha}\notin k$ , is empty. Thus, as in the case of classical varieties, one first wants to figure things out over an algebraic closure and then ‘descend’ back down.

Now, since $k$ is complete we know that there is a unique extension of the absolute value on $k$ to the field $\overline{k}/k$ (e.g. see [Iya, Chapter II Theorem 4.4] or [Bos, Appendix A Theorem 3]). We can thus imagine that, at least over $\overline{k}$ , the ‘closed unit polydisk’ has a clear meaing. Namely, let us make the following definition:

$\mathbb{B}^n_k(\overline{k}):=\left\{(x_1,\ldots,x_n)\in \overline{k}^n:|x_i|\leqslant 1\right\}$

The question then becomes the following: how do we descend back down over $k$ ?

Well, if $X$ is a finite type $k$ -scheme then we can recover $X$ (or, at least, the closed points of $X$ ) as the quotient $X(\overline{k})/\mathrm{Aut}(\overline{k}/k)$ (e.g. see [GW, Proposition 5.4]). This suggests, perhaps, that we can make the definition

$\mathbb{B}^n_k:=\mathbb{B}^n_k(\overline{k})/\mathrm{Aut}(\overline{k}/k)$

Of course, this is a bit unwieldy, and so we’d like to give an alternative description of the set $\mathbb{B}^n_k$ .

If we were back in the situation of a finite type affine $k$ -scheme $X=\mathrm{Spec}(A)$ then it is true that the closed points of $X$ are in bijection with $X(\overline{k})/\mathrm{Aut}(\overline{k}/k)$ but we can also describe the closed points of $X$ as $\mathrm{Max}(A)$ —-the set of maximal ideals of $A$ where $A$ is thought of as the ring of functions on $X$ . Maybe something similarly can be done for $\mathbb{B}^n_k$ .

To begin to do this we need to answer the question “what should the ring of analytic functions on the closed unit polydisk over $k$ disk be?” Well the only reasonable possible answer is

$\begin{aligned} k\langle t_1,\ldots,t_n\rangle &= \left\{f(t_1,\ldots,t_n)\in k[[t_1,\ldots,t_n]]: f(t_1,\ldots,t_n)\text{ converges everywhere on }\mathbb{B}^n_k(\overline{k})\right\}\\ &=\left\{\sum_{I\in\mathbb{N}^n}^\infty a_I t^I \in k[[t_1,\ldots,t_n]]: \lim_{|I|\to\infty}|a_I|=0\right\}\end{aligned}$

where $|(i_1,\ldots,i_n)|=i_1+\cdots+i_n$ . Thus, a naive hope would be then that the natural map

$\mathbb{B}^n_k(\overline{k})/\mathrm{Aut}(\overline{k}/k)\to \mathrm{Max}(k\langle t_1,\ldots,t_n\rangle)\qquad (1)$

(which sends $\alpha\in \mathbb{B}_k(\overline{k})$ to the kernel of the evaluation-at- $\alpha$ map $k\langle t_1,\ldots,t_n\rangle\to\overline{k}$ ) is a bijection. This, in fact, is the case (e.g. see [BGR, §7.1.1 Proposition 1]).

So, we have that $\mathbb{B}^n_k$ is describable, at least as a set, in algebraic terms—it is $\mathrm{Max}(k\langle t_1,\ldots,t_n\rangle)$ . But, of course, we would like to have $\mathbb{B}^n_k$ as more than just as a set—we’d like to have it as some sort of locally ringed space (the correct setting to do ‘modern geometry’).

To this end, let us note that $\mathbb{B}^n_k$ has a topology coming from $(1)$ since $\mathbb{B}^n_k(\overline{k})$ itself has a natural topology—that induced by a basis of open subsets of the form

$\{x\in \mathbb{B}^n_k(\overline{k}): |f(x)|\leqslant \varepsilon \}$

for $\varepsilon$ a positive real number and $f(t_1,\ldots,t_n)\in k\langle t_1,\ldots,t_n\rangle$ . This (quotient) topology on $\mathbb{B}_k$ is called the canonical topology. Note that in the canonical topology there are many other open sets not of the above form—you can have multiple functions defining your inequalities, non-vanishing loci, strict inequality, ‘circles’ (e.g. see [Bos, §3.3]).

Unfortunately, the canonical topology is far too fine. The classical example is to note that $\mathbb{B}_k$ has a natural open cover as the $\mathrm{Aut}(\overline{k}/k)$ -orbits of the following decomposition of $\mathbb{B}_k(\overline{k})$

$\displaystyle \mathbb{B}_k(\overline{k})=\underbrace{\bigcup_{n\geqslant 1}\left\{x\in \overline{k}:|x|\leqslant |\pi|^{\frac{1}{n}}\right\}}_{\mathbb{D}_k(\overline{k})}\sqcup \underbrace{\{x\in \overline{k}:|x|=1\}}_{\mathbb{T}_k(\overline{k})}$

where $\mathbb{D}_k$ is the open disk over $k$ and $\mathbb{T}_k$ is the unit circle (or analytic $1$ -torus). This decomposition shows that with the canonical topology $\mathbb{B}_k$ is neither connected nor quasi-compact in contrast to our stated desires. Another way to think about this is that the sheaf theory doesn’t work for this cover. Namely, again we have that

$\mathbb{B}_k(\overline{k})=\mathbb{D}_k(\overline{k})\sqcup \mathbb{T}_k(\overline{k})$

but certainly one can check that

$\mathcal{O}(\mathbb{B}_k)\ne \mathcal{O}(\mathbb{D}_k)\times \mathcal{O}(\mathbb{T}_k)$

(where $\mathcal{O}(-)$ is something like the ‘convergent power series on $-$ ‘).

Step 2

Before we discuss the fix to this topological problem we actually first discuss the analogue of Step 2 in the theory of rigid spaces. Namely, if we are to think of $\mathbb{B}^n_k$ as $\mathrm{Max}(k\langle t_1,\ldots,t_n\rangle)$ then the analogue of Step 2 should be the following.

Let us first define the analogue of finite type $k$ -algebras. Namely, let $A$ be a $k$ -algebra. We say that $A$ is an affinoid $k$ -algebra if it is a quotient of $k\langle t_1,\ldots,t_n\rangle$ for some $n\geqslant 0$ .

The following properties of affinoid algebras show that they are somewhat similar to polynomial rings:

Lemma 2: Let $A$ be an affinoid algebra. Then:

Any choice of surjection $k\langle t_1,\ldots,t_n\rangle\to A$ defines a quotient topology on $A$ (using the norm $\displaystyle \left|\sum_I a_I t^I\right|:=\sup|a_I|$ ). This topology is actually independent of presentation.

The ring $A$ is Noetherian.

The ring $A$ is Jacobson.

A prime ideal $\mathfrak{p}$ of $A$ is maximal if and only if $A/\mathfrak{p}$ is finite-dimensional over $k$ .

Every ideal of $A$ is closed (in the topology from 1.).

Every homomorphism of $k$ -algebras between affinoid $k$ -algebras is continuous.

Proof: All of these are contained [Tia, Chapter 1] and mostly in [Tia, §1.3]. $\blacksquare$

Now, for any ideal $I\subseteq k\langle t_1,\ldots,t_n\rangle$ we get the subset

$V(I):=\mathrm{Max}(k\langle t_1,\ldots,t_n\rangle/I)\cong \{\mathfrak{m}\in\mathbb{B}_k^n:\mathfrak{m}\supseteq I\}\subseteq \mathbb{B}^n_k$

Of course, one can easily check that we can also identify $V(I)$ as the quotient by $\mathrm{Aut}(\overline{k}/k)$ of the set

$V(I)(\overline{k}):=\{(x_1,\ldots,x_n)\in \mathbb{B}^n_k(\overline{k}):f(x_1,\ldots,x_n)=0\text{ for all }f\in I\}$

In particular, for an affinoid algebra $A$ the choice of a surjection $k\langle t_1,\ldots,t_n\rangle$ identifies $\mathrm{Max}(A)$ concretely in terms of vanishing loci in $\mathbb{B}^n_k(\overline{k})$ and thus endows it with a canonical topology.

Note that by Lemma 2 part 4. that if one has a ring map

$\varphi:A\to B$

of affinoid algebras it induces a map

$\mathrm{Max}(\varphi):\mathrm{Max}(B)\to\mathrm{Max}(A)$

which one can show is continuous with the canonical topology for any presentation of $A$ and $B$ as quotients of some $k\langle t_1,\ldots,t_n\rangle$ (e.g. see [Bos, §3.1, Proposition 6]). In particular, we see that the canonical topology on $\mathrm{Max}(A)$ doesn’t depend on the presentation of $A$ . We call sets of the form $\mathrm{Max}(A)$ affinoid sets (it will be these sets with more structure that are really the affinoid spaces which form the building blocks of rigid analytic varieties).

Moreover, note that a presentation allows us to easily talk about inequalities in an affinoid set. Namely, if we realize $A=k\langle t_1,\ldots,t_n\rangle/I$ then we get an embedding

$\mathrm{Max}(A)\hookrightarrow \mathbb{B}^n_k$

and in particular for a set elements $f_1,\ldots,f_m,g\in A$ we can make senset of the set

$\left\{x\in\mathrm{Max}(A):|f_i(x)|\leqslant |g(x)|\right\}$

Indeed, we can think of the set $\{f_1,\ldots,f_m,g\}$ as a set of functions in $k\langle t_1,\ldots,t_n\rangle$ and take the $\mathrm{Aut}(\overline{k}/k)$ -orbit of the set

$\left\{(x_1,\ldots,x_n)\in \mathbb{B}^n_k(\overline{k}): \begin{aligned}(1)&\quad f(x_1,\ldots,x_n)=0\text{ for all }f\in I\\ (2)&\quad f_i(x_1,\ldots,x_n)\leqslant g(x_1,\ldots,x_n)\text{ for all }i\end{aligned}\right\}\quad (2)$

which is clearly independent of the lifts of $f_1,\ldots,f_m,g\in A$ along the surjection $k\langle t_1,\ldots,t_n\rangle\to A$ .

But, there is an instrinic way to describe these sets as well. Namely, for any $x=\mathfrak{m}\in\mathrm{Max}(A)$ we know that (again by [Bos, §3.1 Proposition 4]) $A/\mathfrak{m}$ is a finite extension of $k$ and therefore carries a unique absolute value extending that of $k$ . One can then define $|f(x)|$ to be the real number given by evaluating this absolute on $A/\mathfrak{m}$ on the element $f\mod \mathfrak{m}$ . One can then check that the set from $(2)$ agrees with the set

$\{x\in\mathrm{Max}(A):|f_i(x)|\leqslant |g(x)|\text{ for all }i\}$

(e.g. see [BGR, §7.1.4 Lemma 2]).

Step 1 and Step 2 (cont.)

Of course, the canonical topology on $\mathrm{Max}(A)$ is too fine as in the special case of $\mathbb{B}^n_k$ . So, let’s return to trying to fix this topology or, what amounts to the same thing geometrically, fixing the sheaf theory. By reexamining the classical the classical theory of the structure sheaf $\mathcal{O}_{\mathrm{Spec}(A)}$ on an affine scheme $\mathrm{Spec}(A)$ Tate was able to see how to fix the sheaf theory on $\mathbb{B}^n_k$ . Namely, how does the structure sheaf $\mathcal{O}_{\mathrm{Spec}(A)}$ work? Well, one begins by working with the distinguished base $\{D(f)\}$ of $\mathrm{Spec}(A)$ (e.g. see [Vak,§2.5] and [Vak, Exercise 3.5.A]) and defining a presheaf on this base as follows:

$\mathcal{O}_{\mathrm{Spec}(A)}(D(f)):=A_f$

One then shows that this is a sheaf on a base (e.g. see [Vak, Theorem 4.1.2]). A pivotal step in this procedure is essentially to use the quasi-compactness of $\mathrm{Spec}(A)$ to reduce to showing that a finite distinguished open cover $\{D(f_i)\}$ satisfies the sheaf property.

Tate then realized that he should figure out what the analogue of a ‘distinguished open’ $D(f)\subseteq \mathrm{Spec}(A)$ inside of $\mathbb{B}^n_k$ is and try to prove that a finite distinguished cover satisfies the sheaf axiom. The analogue of a distinguished open on an affinoid $\mathrm{Max}(A)$ is a so-called rational domain. Namely, for elements $g,f_1,\ldots,f_m\in A$ such that $(g,f_1,\ldots,f_m)$ is the unit ideal one defines the associated rational domain as follows:

$\displaystyle U\left(\frac{f_1,\ldots,f_m}{g}\right)=\{x\in \mathrm{Max}(A):|f_i(x)|\leqslant |g(x)|\}$

which one can pretty easily check are open for the canonical topology on $\mathrm{Max}(A)$ .

Now that we have singled out the correct analogue of $D(f)$ for $\mathrm{Max}(A)$ we want to say what the value of our presheaf $\mathcal{O}_{\mathrm{Max}(A)}$ on these distinguished opens is. The guess is quite clear: the ring $\displaystyle \mathcal{O}_{\mathrm{Max}(A)}\left(U\left(\frac{f_1,\ldots,f_m}{g}\right)\right)$ should be the ring of analytic functions on $\mathrm{Max}(A)$ that are convergent on $\displaystyle U\left(\frac{f_1,\ldots,f_m}{g}\right)$ . Namely, we could embed $\mathrm{Max}(A)$ inside of $\mathbb{B}^n_k$ and say that $\displaystyle \mathcal{O}_{\mathrm{Max}(A)}\left(U\left(\frac{f_1,\ldots,f_m}{g}\right)\right)$ is

$\displaystyle \left\{f\mid_{\mathrm{Max}(A)}:f(t_1,\ldots,t_n)\in k[[t_1,\ldots,t_n]]\text{ which converges everywhere on }U\left(\frac{f_1,\ldots,f_m}{g}\right)\right\}$

(where we have implicitly chosen liftings of $f_1,\ldots,f_m,g$ to $k\langle t_1,\ldots,t_n\rangle$ but it’s clearly independent of such liftings).

But, a coordinate-free (but still a priori dependent on the choice of the set $f_1,\ldots,f_m,g$ ) way to describe this ring is as follows:

$\displaystyle \mathcal{O}_{\mathrm{Max}(A)}\left(U\left(\frac{f_1,\ldots,f_m}{g}\right)\right):=A\langle s_1,\ldots,s_m\rangle/(f_1 s_1-g,\ldots,f_ms_m-g)$

where

$\displaystyle A\langle s_1,\ldots,s_m\rangle := \left\{\sum_{J\in\mathbb{N}^m}a_J s^J\in A[[s_1,\ldots,s_m]]: \lim_{|J|\to\infty}a_J=0\right\}$

where the topology we are using on $A$ (as in Lemma 2 part 1.). In particular, note that $\displaystyle \mathcal{O}_{\mathrm{Max}(A)}\left(U\left(\frac{f_1,\ldots,f_m}{g}\right)\right)$ is still an affinoid $k$ -algebra.

Remark 3: Note that this ring enlarges $k\langle t_1,\ldots,t_n\rangle$ in more ways than adding in inverses. Namely, in normal algebraic geometry essentially if you have an open subset $U\subseteq X$ (let’s say of integral schemes for simplicity) the map $\mathcal{O}(X)\to \mathcal{O}(U)$ allows one to view $\mathcal{O}(U)$ as an enlargement of $\mathcal{O}(X)$ by algebraically adjoining inverses of functions not invertible on $X$ but invertible on $U$ —a sort of generalized localization. In the rigid setting there is not only localization but also completion. The result is that one doesn’t just algebraically adjoin new things but also forces new series to be convergent.

For example $\displaystyle U\left(\frac{t}{\pi}\right)\subseteq \mathbb{B}_k$ has ring of sections $\displaystyle k\langle t\rangle\left\langle\frac{t}{\pi}\right\rangle$ . Now, algebraically adjoining $\displaystyle \frac{t}{\pi}$ does noting (it’s already in the ring) but this ‘completed adjoinment’ forces certain power series in $\displaystyle \frac{t}{\pi}$ to converge—any series of the form $\displaystyle \sum_{i=0}a_i \left(\frac{t}{\pi}\right)^i$ with $\lim a_i=0$ converges in $\displaystyle k\langle t\rangle\left\langle\frac{t}{\pi}\right\rangle$ but needn’t converge in $k\langle t\rangle$ . For a specific example one has that

$\displaystyle \sum_{i=0}^\infty \pi^i \left(\frac{t}{\pi}\right)^i=\sum_{i=0}^\infty t^i$

is in $\displaystyle k\langle t\rangle\left\langle\frac{t}{\pi}\right\rangle$ but not in $k\langle t\rangle$ . $\blacklozenge$

Finally, one can show that for an open subset $U$ of $\mathrm{Max}(A)$ which is a rational domain—which can be presented as $\displaystyle U\left(\frac{f_1,\ldots,f_m}{g}\right)$ for some set $f_1,\ldots,f_m,g\in A$ generating the unit ideal—that the ring $\mathcal{O}_{\mathrm{Max}(A)}(U)$ doesn’t depend on the choice of presentation (i.e. on the set $f_1,\ldots,f_m,g$ )—see [Tia, Example 1.5.12 part 3.]).

We then have the following theorem of Tate:

Theorem 4(Tate): Let $A$ be an affinoid algebra. Then, for any rational domain $U\subseteq \mathrm{Max}(A)$ and any finite covering $\{U_1,\ldots,U_\ell\}$ of $U$ by rational domains (cf. [Bos, §3.3 Proposition 17]) the sequence

$\displaystyle 0\to\mathcal{O}_{\mathrm{Max}(A)}(U)\to \prod_i \mathcal{O}_{\mathrm{Max}}(A_i)\to \prod_{i,j}\mathcal{O}_{\mathrm{Max}(A)}(U_i\cap U_j)$

is exact.

In other words, the presheaf $\mathcal{O}_{\mathrm{Max}(A)}$ is a sheaf with respect to rational domains and finite coverings by rational domains. The idea of proof is to do explicit computations much as in the case of [Vak, Theorem 4.1.2] (for a full proof see the references later on).

In fact, two stronger versions of this theorem can be shown to hold which are often stated instead of Theorem 4. Namely, one can replace rational domains and finite rational coverings by affinoid domains (resp. admissible opens) and finite affinoid coverings (resp. admissible coverings).

The idea of an affinoid subdomain is that it’s basically the analogue of an ‘affine open subscheme’ in the rigid geometry setting. In particular, using this post one is tempted to define a subset $U\subseteq\mathrm{Max}(A)$ to be an affinoid domain if there exists an affinoid algebra $B$ and a map of $k$ -algebras $j^\ast:B\to A$ such that the induced map $j:\mathrm{Max}(B)\to\mathrm{Max}(A)$ has image $U$ and for which $j$ is universal for such a property: if $h^\ast:A\to C$ is a $k$ -algebra map of affinoid algebras such that $h:\mathrm{Max}(C)\to\mathrm{Max}(A)$ has image landing in $U$ then $h^\ast$ unique factors through $j^\ast$ . One can show that such $U$ are open for the canonical topology (see [Tia, Corollary 1.5.22]) and that the $k$ -algebra $B$ is unique up to unique isomorphism.

As an example, one can show that rational domains are affinoid domains (e.g. see [Tia, Example 1.5.12 part 3.]). In fact, the super important Gerritzen–Grauert theorem gives a partial converse to this statement: every affinoid domain is a finite union of rational domains. Now, given the unicity of $B$ (as in the last paragraph) one can associate to any affinoid domain $U\subseteq \mathrm{Max}(A)$ the ring $\mathcal{O}_{\mathrm{Max}(A)}(U):=B$ .

One then has the following strengthening of Theorem 4:

Theorem 5: Let $A$ be an affinoid algebra. Then, for any affinoid domain $U\subseteq \mathrm{Max}(A)$ and any finite covering $\{U_1,\ldots,U_\ell\}$ of $U$ by affinoid domains (cf. [Bos, §3.3 Proposition 12]) the sequence

$\displaystyle 0\to\mathcal{O}_{\mathrm{Max}(A)}(U)\to \prod_i \mathcal{O}_{\mathrm{Max}}(A_i)\to \prod_{i,j}\mathcal{O}_{\mathrm{Max}(A)}(U_i\cap U_j)$

is exact.

Proof: See [Tia, Theorem 1.6.3]. The idea is to use the Gerritzen–Grauert theorem to reduce this to rational domains and rational covers, then even simpler types of opens and covers and finally show the sheaf condition explicitly by hand. $\blacksquare$

The final strengthening allows one to replace affinoid domains $U\subseteq\mathrm{Max}(A)$ , which (again) one should think of as just open affinoid subspaces, with certain open subspaces and certain covers. The idea is that whatever general open subset $U$ of $\mathrm{Max}(A)$ we encounter, one shouldn’t be able to produce using $U$ another affinoid $\mathrm{Max}(B)$ (not necessarily an affinoid open in $\mathrm{Max}(A)$ and an affinoid cover of $\mathrm{Max}(B)$ which does not have a finite subcover—for this is precisely the case in which we have constructed our theory thusfar to avoid. How does one make this rigorous?

Well, let us call an open subset $U\subseteq\mathrm{Max}(A)$ admissible if there exists an affinoid (in $\mathrm{Max}(A)$ ) cover (possibly infinite) $\{U_i\}$ of $U$ such that for any map of affinoid $k$ -algebras $\varphi^\ast:A\to B$ such that the induced map $\varphi:\mathrm{Max}(B)\to\mathrm{Max}(A)$ has $\varphi(\mathrm{Max}(B))\subseteq U$ one has that the affinoid cover $\{\varphi^{-1}(U_i)\}$ of $\mathrm{Max}(B)$ has a refinement (cf. [Bos, Pg. 83]) by a finite affinoid open cover. Similarly, an admissible covering of the admissible open $U$ is a collection of admissible open subsets $\{V_j\}$ of $\mathrm{Max}(A)$ which cover $U$ and have the property that for any $k$ -algebra map of $k$ -affinoids $\varphi^\ast:A\to B$ such that $\varphi(\mathrm{Max}(B))\subseteq U$ one has that the cover $\{\varphi^{-1}(V_j)\}$ has a refinement by a finite affinoid open cover.

One should interpret non-admissible covers of an affinoid $\mathrm{Max}(A)$ as ‘bad’ and, in particular, should be disregarded.

Example 6: The cover the open unit disk $\mathbb{D}_k$ is an admissible open subset of $\mathbb{B}_k$ (e.g. see [Tia, Example 2.1.6]) and $\mathbb{T}_k$ is admissible (it’s even rational—it’s $\displaystyle U\left(\frac{1}{t}\right)$ ) but the cover $\{\mathbb{D}_k,\mathbb{T}_k\}$ is not admissible (e.g. see [Con1, Example 2.2.8]). We will later give a good interpretation for this in terms of adic spaces. $\blacklozenge$

We then have the following general strengthening of Theorem 4 and Theorem 5:

Theorem 7:Let $A$ be an affinoid algebra. Then, for any admissible open $U\subseteq \mathrm{Max}(A)$ and any admissible covering $\{V_i\}$ of $U$ by admissible opens the sequence

$\displaystyle 0\to\mathcal{O}_{\mathrm{Max}(A)}(U)\to \prod_i \mathcal{O}_{\mathrm{Max}}(V_i)\to \prod_{i,j}\mathcal{O}_{\mathrm{Max}(A)}(V_i\cap V_j)$

is exact.

Step 3

Now, we have singled out three (increasingly general) situations where our sheaf theory has worked:

It works for rational domains in $\mathrm{Max}(A)$ and finite rational covers.
It works for affinoid domains in $\mathrm{Max}(A)$ and finite affinoid covers (this is the ‘obvious’ version of 1.—it’s just easier to work with 1. than general finite affinoid covers)
It works for admissible opens in $\mathrm{Max}(A)$ and admissible covers (this is the finest replacement $G$ -topology [see below] which ‘acts the same’ as the $G$ -topology from 2.—see [BGR, §9.1.2] for more details).

In all three cases we see that we are in the situation of having some set of distinguished open subsets of $\mathrm{Max}(A)$ (rational domains, affinoid domains, admissible opens) and for each distinguished open subset a ‘permissible’ set of covers of that distinguished open subset by distinguished open subsets (finite rational covers, finite affinoid covers, admissible covers). Now, none of these are an acutal topology on $\mathrm{Max}(A)$ (e.g. they aren’t closed under arbitrary unions) but they act similarly enough like a topology to make sheaf theory work. In fact, they are an example of a $G$ -topology.

Roughly, a $G$ topology on a topological space $X$ is

A (usually proper) collection $\mathrm{Perm}(X)$ of open subsets of $X$ ‘the permissible opens’.
For each permissible open subset $U\in\mathrm{Perm} (X)$ a set $\mathrm{Cov}(U)$ of open covers of $U$ by permissible opens—‘the permissible covers’ (NB: not every open cover by permissible opens is assumed permissible—just a subset of such covers).

One needs to assume that the above colletions have reasonable enough properties to talk about sheaf theory (for a rigorous definition see [Bos, §5.1] and [Bos, §5.2]). Namely, one calls a contravariant functor

$\mathcal{F}:\left\{\text{Permissible opens}\right\}\to\mathsf{Ab}$

(where $\mathsf{Ab}$ is the category of abelian groups and the set of permissible opens is considered a category where the maps are the inclusions of permissible open sets) an abelian presheaf on the $G$ -space $X$ . One calls an abelian presheaf $\mathcal{F}$ on the $G$ -space $X$ an abelian sheaf if for all $U\in\mathrm{Perm}(X)$ and all $\{U_i\}\in\mathrm{Cov}(U)$ the presheaf $\mathcal{F}$ satisfies the sheaf conditions for the cover— in other words the sequence

$\displaystyle 0\to\mathcal{F}(U)\to \prod_i \mathcal{F}(U_i)\to \prod_{i,j}\mathcal{F}(U_i\cap U_j)$

is exact. Of course, one can define sheaves with values in any reasonable category (e.g. see [Stacks, 00VL] or [Bos, §5.1] and [Bos, §5.2]).

Remark 8: The reason for the name ‘ $G$ -topology’ is that a $G$ -topology on $X$ is, essentially, a Grothendieck subtopology of $\mathrm{Open}(X)$ for those familiar with Grothendieck topologies (e.g. from étale cohomology). This allows one to use all the theory one has learned from the theory of cohomology on sites etc. (e.g. [Stacks, 00UZ] and [Stacks, 01FQ]) $\blacklozenge$

Note that what we are calling here ‘permissible opens’ and ‘permissible covers’ is usually called ‘admissible opens’ and ‘admissible covers’, but so as to not confuse these with our explicit admissible covers for affinoid sets we have elected to use this non-standard terminology.

Given a $G$ -space $X$ and an abelian sheaf $\mathcal{F}$ on $X$ one can define the stalk of $\mathcal{F}$ at a point $x\in X$ . Namely, one can define

$\displaystyle \mathcal{F}_x:=\varinjlim_{U\in N(x)}\mathcal{F}(U)$

where $N(x)$ is the set of permissible open subsets of $X$ containing $x$ .

A locally ringed $G$ -space is a $G$ -space $X$ with a sheaf $\mathcal{O}_X$ of rings whose stalks are local rings. If, in addition, $\mathcal{O}_X$ is a sheaf of $k$ -algebras we call $X$ a locally ringed $G$ -spaces over $k$ . A morphism of locally ringed $G$ -spaces and/or locally locally ringed $G$ – space over $k$ is the obvious one (e.g. again see [Stacks,00VL] or [Bos, §5.1] and [Bos, §5.2]).

Now, from the discussion in the previous subsections we see that for every affinoid algebra $A$ one gets two (major) locally ringed $G$ -space over $k$ given as the pair

$\mathrm{Sp}(A):=(\mathrm{Max}(A),\mathcal{O}_{\mathrm{Max}(A)})$

where the permissible opens are the admissible opens (resp. affinoid domains), and the permissible covers are the admissible ones (resp. finite affinoid covers). We call this the strong (resp. weak) $G$ -topology on $\mathrm{Max}(A)$ . Unless specified otherwise, the $G$ -topology on $\mathrm{Max}(A)$ will always taken to be the strong one. We call a locally ringed $G$ -space over $k$ of the form $(\mathrm{Max}(A),\mathcal{O}_{\mathrm{Max}(A)})$ (with the strong $G$ -topology) an affinoid space over $k$ . While $\mathrm{Sp}(A)$ is really a pair, we shall, in practice conflate $\mathrm{Sp}(A)$ and $\mathrm{Max}(A)$ .

This pair satisfied the expected property:

Lemma 9: Let $A$ and $B$ be affinoid $k$ -algebras. Then, the map

$\mathrm{Hom}(\mathrm{Sp}(B),\mathrm{Sp}(A))\to \mathrm{Hom}_k(A,B)$

given by the map on global sections is an isomorphism. In other words the functor

$\mathrm{Sp}:\left\{\begin{matrix}\text{Affinoid algebras}\\ \text{over }k\end{matrix}\right\}\to \left\{\begin{matrix}\text{Locally ringed }G-\text{spaces}\\ \text{over }k\end{matrix}\right\}$

is fully faithful.

Proof: See [Bos, §5.3 Corollary 7]. $\blacksquare$

Now, given $(X,\mathcal{O}_X)$ any locally ringed $G$ -space over $k$ and any open permissible subset $U\subseteq X$ one naturally gets a $G$ -topology on $U$ (e.g. $\mathrm{Perm}(U)$ is the set of elements of $\mathrm{Perm}(X)$ contained in $U$ ) and one can restrict $\mathcal{O}_X$ to get a sheaf $\mathcal{O}_U:=\mathcal{O}_X\mid_U$ on $U$ so that the pair $(U,\mathcal{O}_U)$ is also a locally ringed $G$ -space over $k$ .

So, with all of this setup we can finally defined rigid analytic varieties a la Tate. Namely, a rigid analytic variety is a pair $(X,\mathcal{O}_X)$ which is a locally ringed $G$ -space over $k$ such that there is a permissible open cover $\{U_i\}$ of $X$ such that $(U_i,\mathcal{O}_{U_i})$ is isomorphic to an affinoid space. A morphism of rigid spaces is a morphism of locally ringed $G$ -spaces over $k$ .

One has a natural generalization of Lemma 9:

Lemma 10: Let $X$ and $Y$ be rigid spaces with $Y$ affinoid. Then, the natural map

$\mathrm{Hom}(X,Y)\to \mathrm{Hom}_k(\mathcal{O}(Y),\mathcal{O}(X))$

is a bijection.

Proof: See [Bos, §5.3 Corollary 7]. $\blacksquare$

Let us finally note that the set of affinoid open subspaces of a rigid space $X$ form a distinguished base for $X$ (in the sense of [Vak, §2.5]) and thus if one wants to define a sheaf on $X$ it suffices to define a sheaf on the distinguished base of affinoid opens (e.g. by [Vak, Theorem 2.5.1]).

The examples

Now with all of this setup we can finally discuss some simple examples of rigid spaces. We shall be introducing some more theory along the way.

The closed unit disk

We start with the basic object which we have already seen many times above. Namely, the closed unit disk (or closed unit ball) over $k$ is the rigid analytic variety $\mathbb{B}_k:=\mathrm{Sp}(k\langle t\rangle)$ . Let us start to prove some of the basic properties of $\mathbb{B}_k$ .

Topological properties

Perhaps the first thing to check is that $\mathbb{B}_k$ , as we envisioned it, is quasi-compact and connected. Of course, since $\mathbb{B}_k$ is actually a $G$ -space (in particular not a topological space) we need to define what this means. But, of course, it’s the obvious thing. Namely a $G$ -space $X$ is quasi-compact if every permissible cover of $X$ has a finite refinement. And, a $G$ -space $X$ is connected if there does not exist a permissible cover $\{U,V\}$ of $X$ such that $U\cap V$ is empty and $U,V\ne\varnothing$

To prove that $\mathbb{B}_k$ is quasi-compact and connected we make the following general observation:

Proposition 11: Let $A$ be an affinoid $k$ -algebra. Then, $\mathrm{Sp}(A)$ is quasi-compact. Moreover, $\mathrm{Sp}(A)$ is connected if and only if $A$ has no non-trivial idempotents.

Proof: We tailor build admissible covers so to force the quasi-compactness. Namely, if $\{U_i\}$ is an admissible cover of $\mathrm{Sp}(A)$ then letting $\varphi^\ast:A\to A$ be the identity map (in the definition of admissible) one sees by definition that $\{\varphi^{-1}(U_i)\}=\{U_i\}$ must admit a finite affinoid refinement.

To see the second claim we proceed as usual. If $X=U\sqcup V$ with $\{U,V\}$ admissible then $\mathcal{O}(X)=\mathcal{O}(U)\times \mathcal{O}(V)$ and so $\mathcal{O}(X)$ has non-trivial idempotents as soon as $\mathcal{O}(U)$ and $\mathcal{O}(V)$ are not the zero ring. To see this note that if $U$ is a non-empty rigid space then $\mathcal{O}(U)$ is non-zero. Indeed, let $x\in U$ and let $W$ be an affinoid neighborhood of $x$ . Then, we have a ring map $\mathcal{O}_U(U)\to\mathcal{O}_W(W)$ and, in particular, if $\mathcal{O}_U(U)$ is the zero ring this forces $\mathcal{O}_W(W)$ to be the zero ring which forces $W=\mathrm{Max}(\mathcal{O}_W(W))$ to be empty, which is a contradiction.

Conversely, suppose that $\mathcal{O}(X)$ has non-trivial idempotents so that $\mathcal{O}(X)=R\times S$ with $R,S$ not zero rings. Let $e:=(1,0)$ and $f:=(0,1)$ . Then I leave it to the reader to check that the non-vanishing locus of $e$ and $f$ , called $U$ and $V$ , are open, non-intersecting, and that $\{U,V\}$ is an admissible cover of $X$ . $\blacksquare$

That said, let’s notice our first descrepancy with the scheme picture (although it is consonant with the classical picture of varieties). Namely, recall that if $A$ is an integral domain that $\mathrm{Spec}(A)$ is more than just connected it’s actually irreducible. This irreducibility, and the existence of generic points, are a key component of what makes scheme-theoretic algebraic geometry tick. This completely fails for objects in the rigid setting. Namely, let us make the following non-standard definition. Namely, let us say that a $G$ -space is irreducible if for all admissible opens $U,V\in\mathrm{Perm}(X)$ we have that $U\cap V\ne\varnothing$ . Essentially no rigid space is irreducible.

For example, certainly $\mathbb{B}_k$ is not irreducible since, for example,

$\mathbb{D}_k=\{x\in \mathbb{B}_k:|x|<1\}$

and

$\mathbb{T}_k=\{x\in\mathbb{B}_k:|x|=1\}$

are admissible opens (we leave this to the reader, or see [Con1, Example 2.2.8]) which intersect disjointly. If one wants to soup this up to an example consisting only of affinoid opens one can take two non-equal points $x_0,x_1\in\mathbb{B}_k(k)$ and let $|x_0-x_1|=\varepsilon>0$ . Then,

$\displaystyle U_i:=\left\{x\in\mathbb{B}_k:|x-x_i|\leqslant \frac{\varepsilon}{2}\right\}$

are disjoint admissible opens. Thus, $\mathbb{B}_k$ is far from being ‘irreducible’.

Remark 12: One should take this reducibility of (most) rigid spaces as both a blessing and curse. Namely, the fact that our spaces are reducible means that they are closer to the normal topological spaces one deals with in complex geometry and therefore we can often times perform natural operations in the rigid setting which are impossible to perform in the scheme setting (e.g. see the discussion on Tate elliptic curves below).

But, on the other hand, we are now robbed of one of the most powerful tools at the disposal of algebraic geometers: the ubiquitous generic point. Since most reasonable schemes we work with are irreducible they have generic points and, in particular, one can often use this generic point to simplify a huge amount of technical assumptions. I didn’t realize how important generic points were to my understanding of scheme theory until I tried to do rigid geometry where such points were suddenly missing.

Let us also note here by that ‘irreducible’ we mean irreducible in the naive sense—irreducible as topological spaces. This, of course, is the type of irreducibility we usually mean in the context of schemes. That said, there is a more ‘algebraic’ notion of irreducibility which does work well in the context of rigid spaces. Essentially, one replaces the usual $G$ -topology for a coarser one—the Zariski topology. For this, one can see [Con3]. $\blacklozenge$

Functor of points

Let us now contemplate what the functor of points of $\mathbb{B}_k$ is. Namely, we have the following observation:

Proposition 13: For any rigid space $X$ over $k$ is a natural isomorphism of sets

$\mathrm{Hom}(X,\mathbb{B}_k)\xrightarrow{\approx}\mathcal{O}_X^+(X)$

Before we prove this let us define what $\mathcal{O}_X^+$ is. This is a presheaf of $\mathcal{O}_k$ -algebras on $X$ defined as follows. For an affinoid open $\mathrm{Sp}(A)\subseteq X$ we define

$\mathcal{O}_X^+(\mathrm{Sp}(A)):=A^\circ$

where $A^\circ$ is the ring of power bounded elements of $A$ (i.e. those elements $a\in A$ such that the set $\{a^n\}$ is bounded—see [Bos, §3.1 Theorem 17], [Tia, Proposition 1.4.13], or [BGR, §1.2.5]). Note that this is a sheaf with respect to the distinguished base of affininoid opens (this is easy and left to the reader—use the equivalences given in loc. cit.). Then, as previously remarked, $\mathcal{O}_X^+$ extends uniquely to a sheaf on the $G$ -space $X$ .

One then sees $\mathcal{O}_X^+(X)$ as the global sections of this sheaf. Intuitively one imagines that $\mathcal{O}_X^+(X)$ is the set of elements $\alpha$ of $\mathcal{O}_X(X)$ such that $|\alpha|\leqslant 1$ (e.g. see [Tia, Proposition 1.4.13] for a rigorous statement) from which Proposition 13 then makes intuitive sense.

Finally we can prove Proposition 13:

Proof( Proposition 13): Note that since $\mathrm{Hom}(-,\mathbb{B}_k)$ and $\mathcal{O}_X^+$ are sheaves for the $G$ -topology it suffices to prove this natural isomorphism when $X$ is affinoid, say $X=\mathrm{Sp}(A)$ . Note then that by Lemma 9 there is a natural bijection

$\mathrm{Hom}(\mathrm{Sp}(A),\mathbb{B}_k)=\mathrm{Hom}_k(k\langle t\rangle, A)$

Now, note that since any $k$ -homomorphism $f:k\langle t\rangle\to A$ is continuous (e.g. see Lemma 38) that such a homomorphism is uniquely determined by where it sends $t$ . Namely, if $t\mapsto \alpha\in A$ then

$\begin{aligned}f\left(\sum_i a_i t^i\right) &=f\left(\lim \sum_{i=0}^N a_i t^i\right)\\ &=\lim f\left(\sum_{i=0}^N a_i t^i\right)\\ &= \lim \sum_{i=0}^N a_i f(t)^i\\ &= \sum_{i=0}^\infty a_i \alpha^i\end{aligned}$

Moreover, this also easily shows that $\alpha$ is of the form $f(t)$ if and only if $\displaystyle \sum_{i=0}^\infty a_i \alpha^i$ converges for every sequence $\{a_i\}$ in $k$ such that $\lim a_i=0$ . But it’s easy to see that this is equivalent to $\alpha\in A^\circ$ . The conclusion follows. $\blacksquare$

In particular, let us note that since $\mathcal{O}_X^+(X)$ is a ring (as we leave to the reader to check) one has that $\mathbb{B}_k$ is a ring object in the category of rigid spaces over $k$ . This is perhaps not too surprising since $\mathbb{B}_k$ is meant to be the rigid geometry avatar of $\mathcal{O}_k$ which is itself a ring.

Picard group

Let us also try and say something slightly more interesting about $\mathbb{B}_k$ . Namely, let us define a line bundle on a rigid space $X$ as a sheaf $\mathcal{F}$ of $\mathcal{O}_X$ -modules which locally (for the $G$ -topology) is isomorphic to $\mathcal{O}_X$ (cf. [FdvP, Definition 4.5.1]). As per usual the group of (isomorphism classes of) line bundles on $X$ forms a group under tensor product which is called the Picard group of $X$ and written $\mathrm{Pic}(X)$ . What is the Picard group of $\mathbb{B}_k$ ?

To give us some intuition, let us first try and understand a closely related question over the complex numbers—we will later see this is actually a bit deceiving, but let’s use it anyways:

Example 14: Let $\mathbb{D}_\mathbb{C}$ be the open unit disk thought of as a complex manifold. We claim that $\mathrm{Pic}(\mathbb{D}_\mathbb{C})=0$ . Indeed, we have the exponential sequence

$0\to \underline{2\pi i \mathbb{Z}}\to \mathcal{O}_{\mathbb{D}_\mathbb{C}}\to \mathcal{O}^\times_{\mathbb{D}_{\mathbb{C}}}\to 0$

Taking the long exact sequence in cohomology, using the fact that

$H^i(\mathbb{D}_\mathbb{C},\underline{2\pi i\mathbb{Z}})\cong H^i_\mathrm{sing}(\mathbb{D}_{\mathbb{C}},\mathbb{Z})$

(e.g. see [Wed, Theorem 11.13]), and the fact that $\mathbb{D}_\mathbb{C}$ is contractible we get a natural isomorphism

$H^1(\mathbb{D}_\mathbb{C},\mathcal{O}_{\mathbb{D}_\mathbb{C}})\cong H^1(\mathbb{D}_{\mathbb{C}},\mathcal{O}_{\mathbb{D}_{\mathbb{C}}}^\times)$

and of course we have natural isomorphisms

$\mathrm{Pic}(\mathbb{D}_{\mathbb{C}})\cong H^1(\mathbb{D}_{\mathbb{C}},\mathcal{O}_{\mathbb{D}_{\mathbb{C}}}^\times)$

Thus, it suffices to show that $H^1(\mathbb{D}_{\mathbb{C}},\mathcal{O}_{\mathbb{D}_{\mathbb{C}}})=0$ . This last equality can be shown to be true by the Mittag-Leffler theorem. $\blacklozenge$

So, a guess might be that since the open disk and closed disk are not too far apart that the Picard group of $\mathbb{B}_k$ is also trivial. This, as it turns out, is correct:

Proposition 15: Let $A$ be an affinoid $k$ -algebra. Then, there is a natural isomorphism

$\mathrm{Pic}(\mathrm{Spec}(A))\xrightarrow{\approx}\mathrm{Pic}(\mathrm{Sp}(A))$

given by associating a finitely generated projective $A$ -module $M$ with the sheaf $\widetilde{M}$ on $\mathrm{Sp}(A)$ which associates to an affinoid open $\mathrm{Sp}(B)\subseteq\mathrm{Sp}(A)$ the tensor product $M\otimes_A B$ .

Proof: See [Bos, §6.1 Theorem 4] or [FvdP, Proposition 4.7.2]. $\blacksquare$

In particular, we see that proving that $\mathrm{Pic}(\mathbb{B}_k)=0$ is equivalent to proving that $\mathrm{Pic}(\mathrm{Spec}(k\langle t\rangle))=0$ . But, to do this it suffices (e.g. see [Stacks, 0BCH]) to show that $k\langle t\rangle)$ is a PID. But, this is well-known (e.g. see [Bos, §2.2 Corollary 10]).

Dimension

Defining the dimension for a rigid space $X$ in the classical Tate perspective is necessarily ad-hoc. Namely, one is tempted to try and define it as in the case of schemes—the dimension is the maximal chain of specializations. But, of course, this cannot work well since we have only closed points! One might then try to make an adapatation of what we do in classical variety language (e.g. as in [Har, Chapter 1]) where we use chains of irreducible closed subsets. But, as we saw above (around Remark 12) that irreducible analytic spaces are quite rare. In the Huber and/or Raynaud setting there is a natural fix to these issues, but for now we instead take the following as our definition of dimension. The dimension of a rigid space $X$ is the supremum of $\mathrm{dim}(\mathcal{O}_{X,x})$ as $x$ ranges over the points of $x$ .

So, what is the dimension of $\mathbb{B}_k$ ? Of course the answer is $1$ , but we need to verify this. But, as observed above, since $\mathbb{B}_k$ is a group under addition it suffices to check $\dim\mathcal{O}_{\mathbb{B}_k,0}$ . What is this ring? Well, almost by definition (since closed subdisks around $0$ are clearly a basis for $N(0)$ —cf. our definition of stalk) that

$\begin{aligned}\mathcal{O}_{\mathbb{B}_k,0} &=\varinjlim_{n\to\infty }k\langle t,s\rangle/(\pi^n s-t)\\ &\cong \left\{\sum_{i=0}^\infty a_i t^i \in k[[t]]: \lim_{i\to\infty} |\pi^{ni} a_i|=0\text{ for some }n\geqslant 0\right\}\end{aligned}$

In other words, $\mathcal{O}_{\mathbb{B}_k,0}$ is nothing but the ring of power series convergent in some neighborhood $0$ . This is, by inspection, a DVR. Indeed, it’s easy to see that $(t)$ is the unique maximal ideal of $\mathcal{O}_{\mathbb{B}_k,0}$ (e.g. it’s the same proof as here) from where one can easily deduce the conclusion.

Thus,

$\dim\mathbb{B}_k=\dim \mathcal{O}_{\mathbb{B}_k,0}=1$

as desired.

The punctured closed disk

We now would like to consider the spaces $\mathbb{B}_k-\{p_1,\ldots,p_m\}$ where $p_i$ are any points of $\mathbb{B}_k$ . We call such a space a (several times) punctured disk. For sake of simplicity we assume that $k$ is algebraically closed (or that the points $p_1,\ldots,p_m$ are $k$ -points). This does not change the story in any substantive way, it just allows one to not have to worry about an $L$ -point, for $L/k$ finite, splitting in to several points over $\overline{k}$ .