This is the first in a series of 4 posts whose goal is to briefly introduce rigid geometry with a focus on examples. The ultimate goal is to continue this post (and its sequels) with the reader hopefully having a broad grasp of the rigid geometry of the objects involved.
We shall not go into the motivation for rigid geometry in any 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
which one can read about, for example, in the book [Nee]. Intuitively, this analytification functor takes a -scheme to which is endowed with the natural structure of a -manifold (it’s really only a manifold if 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 -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 and the analytic topology of 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 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 is useful for proving purely geometric results, the passage from something like to via the Lefschetz principle completely forgets arithmetic. Namely, if is a variety over then the variety has an action of the absolute Galois group which is of great interest to number theorists. Such information is completely lost in the passage from to .
To remedy this one might note that , like , 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 -adic analytification functor which takes varieties over and outputs ‘-adic analytic manifolds’. The hope is that such -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 -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 -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 ‘ 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 five (really more than 5, but we won’t mention the other’s here) theories of rigid analytic geometry that we hope to touch on in these pots. 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. 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 should be thought of as the ‘generic fibers’ of formal schemes over . 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.
All of these theories are relatively different in their presentation even though they generally capture, at their heart, the same theory. 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 four perspectives: the classical Tate perspective on rigid spaces, the perspective of Raynaud’s formal fiber, the perspective of Berkovich spaces and Huber’s perspective in terms of adic spaces. We shall briefly mention Fujiwara–Kato’s perspective, though we will not have a full post devoted to it.
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 ), 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.
- 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 —namely spaces locally isomorphic to closed subsets of . 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 to be a non-archimedean complete (but not necessarily locally compact) field with respect to an absolute value . Examples of such fields are
- The field of -adic numbers with the usual -adic absolute value (which we will just usually abbreviate to ), or finite extensions thereof (with the unique extension of ).
- The field which is the completion of the maximal unramified extension of (with respect to the unique extension of ).
- The field which is the completion of (with respect to the unique extension of ).
- The field with the usual -adic absolute value , or a finite extension thereof (with the unique extension of the -adic absolute value).
- The field with the -adic absolute value.
Let us also fix an element of satisfying (such an element is called a pseudo-uniformizer since it plays the role for such that a uniformizer does for discretely valued fields.
The general setup
As mentioned above, Tate’s perspective is the beginning of rigid geometry which, in essence, tries to capture an ‘algebraic theory of -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
we also want to consider opens given by inequalities involving analytic functions
(which, by the magic of non-archimedean theory, are open in the topology given by ) 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 -adic manifolds some sort of surprising things will happen. For example, the closed unit disk over , denoted , 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 is some ring of analytic functions (convergent power series) and, in particular, an integral domain. So, if is to have a sheaf of analytic functions, the fact that is an integral domain actually forces connectedness (e.g. see the argument in [Stacks, Tag00EC]). This is in stark contrast to the naive -adic manifold
which, with its native -adic topology, is neither connected nor quasi-compact.
Now, just as one can imagine building the theory of varieties in three steps:
- (Step 1) Define affine space .
- (Step 2) Define affine varieties as closed subsets .
- (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 now explain this progression roughly.
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 which one imagines as
(we will shorten to ) as our building blocks in Step 1.
Of course, the naive guess for what the closed unit polydisk should be is that it’s, roughly, the set
Of course, if is not algebraically closed this has the exact same pitfalls as if one considers to be the ‘affine line’ over . For example, the closed subspace , where , 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 is complete we know that there is a unique extension of the absolute value on to the field (e.g. see [Iya, Chapter II Theorem 4.4] or [Bos, Appendix A Theorem 3]). We can thus imagine that, at least over , the ‘closed unit polydisk’ has a clear meaing. Namely, let us make the following definition:
The question then becomes the following: how do we descend back down over ?
Well, if is a finite type -scheme then we can recover (or, at least, the closed points of ) as the quotient (e.g. see [GW, Proposition 5.4]). This suggests, perhaps, that we can make the definition
Of course, this is a bit unwieldy, and so we’d like to give an alternative description of the set .
If we were back in the situation of a finite type affine -scheme then it is true that the closed points of are in bijection with but we can also describe the closed points of as —-the set of maximal ideals of where is thought of as the ring of functions on . Maybe something similarly can be done for .
To begin to do this we need to answer the question “what should the ring of analytic functions on the closed unit polydisk over disk be?” Well the only reasonable possible answer is
where . Thus, a naive hope would be then that the natural map
(which sends to the kernel of the evaluation-at- map ) is a bijection. This, in fact, is the case (e.g. see [BGR, §7.1.1 Proposition 1]).
So, we have that is describable, at least as a set, in algebraic terms—it is. But, of course, we would like to have 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 has a topology coming from since itself has a natural topology—that induced by a basis of open subsets of the form
for a positive real number and . This (quotient) topology on 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 has a natural open cover as the -orbits of the following decomposition of
where is the open disk over and is the unit circle (or analytic -torus). This decomposition shows that with the canonical topology 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
but certainly one can check that
(where is something like the ‘convergent power series on ‘).
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 as then the analogue of Step 2 should be the following.
Let us first define the analogue of finite type -algebras. Namely, let be a -algebra. We say that is an affinoid -algebra if it is a quotient of for some .
The following properties of affinoid algebras show that they are somewhat similar to polynomial rings:
Lemma 2: Let be an affinoid algebra. Then:
- Any choice of surjection defines a quotient topology on (using the norm ). This topology is actually independent of presentation.
- The ring is Noetherian.
- The ring is Jacobson.
- A prime ideal of is maximal if and only if is finite-dimensional over .
- Every ideal of is closed (in the topology from 1.).
- Every homomorphism of -algebras between affinoid -algebras is continuous.
Proof: All of these are contained [Tia, Chapter 1] and mostly in [Tia, §1.3].
Now, for any ideal we get the subset
Of course, one can easily check that we can also identify as the quotient by of the set
In particular, for an affinoid algebra the choice of a surjection identifies concretely in terms of vanishing loci in and thus endows it with a canonical topology.
Note that by Lemma 2 part 4. that if one has a ring map
of affinoid algebras it induces a map
which one can show is continuous with the canonical topology for any presentation of and as quotients of some (e.g. see [Bos, §3.1, Proposition 6]). In particular, we see that the canonical topology on doesn’t depend on the presentation of . We call sets of the form 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 then we get an embedding
and in particular for a set elements we can make senset of the set
Indeed, we can think of the set as a set of functions in and take the -orbit of the set
which is clearly independent of the lifts of along the surjection .
But, there is an instrinic way to describe these sets as well. Namely, for any we know that (again by [Bos, §3.1 Proposition 4]) is a finite extension of and therefore carries a unique absolute value extending that of . One can then define to be the real number given by evaluating this absolute on on the element . One can then check that the set from agrees with the set
(e.g. see [BGR, §7.1.4 Lemma 2]).
Step 1 and Step 2 (cont.)
Of course, the canonical topology on is too fine as in the special case of . 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 on an affine scheme Tate was able to see how to fix the sheaf theory on . Namely, how does the structure sheaf work? Well, one begins by working with the distinguished base of (e.g. see [Vak,§2.5] and [Vak, Exercise 3.5.A]) and defining a presheaf on this base as follows:
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 to reduce to showing that a finite distinguished open cover satisfies the sheaf property.
Tate then realized that he should figure out what the analogue of a ‘distinguished open’ inside of is and try to prove that a finite distinguished cover satisfies the sheaf axiom. The analogue of a distinguished open on an affinoid is a so-called rational domain. Namely, for elements such that is the unit ideal one defines the associated rational domain as follows:
which one can pretty easily check are open for the canonical topology on .
Now that we have singled out the correct analogue of for we want to say what the value of our presheaf on these distinguished opens is. The guess is quite clear: the ring should be the ring of analytic functions on that are convergent on . Namely, we could embed inside of and say that is
(where we have implicitly chosen liftings of to but it’s clearly independent of such liftings).
But, a coordinate-free (but still a priori dependent on the choice of the set ) way to describe this ring is as follows:
where the topology we are using on (as in Lemma 2 part 1.). In particular, note that is still an affinoid -algebra.
Remark 3: Note that this ring enlarges in more ways than adding in inverses. Namely, in normal algebraic geometry essentially if you have an open subset (let’s say of integral schemes for simplicity) the map allows one to view as an enlargement of by algebraically adjoining inverses of functions not invertible on but invertible on —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 has ring of sections . Now, algebraically adjoining does noting (it’s already in the ring) but this ‘completed adjoinment’ forces certain power series in to converge—any series of the form with converges in but needn’t converge in . For a specific example one has that
is in but not in .
Finally, one can show that for an open subset of which is a rational domain—which can be presented as for some set generating the unit ideal—that the ring doesn’t depend on the choice of presentation (i.e. on the set )—see [Tia, Example 1.5.12 part 3.]).
We then have the following theorem of Tate:
Theorem 4(Tate): Let be an affinoid algebra. Then, for any rational domain and any finite covering of by rational domains (cf. [Bos, §3.3 Proposition 17]) the sequence
In other words, the presheaf 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 to be an affinoid domain if there exists an affinoid algebra and a map of -algebras such that the induced map has image and for which is universal for such a property: if is a -algebra map of affinoid algebras such that has image landing in then unique factors through . One can show that such are open for the canonical topology (see [Tia, Corollary 1.5.22]) and that the -algebra 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 (as in the last paragraph) one can associate to any affinoid domain the ring .
One then has the following strengthening of Theorem 4:
Theorem 5: Let be an affinoid algebra. Then, for any affinoid domain and any finite covering of by affinoid domains (cf. [Bos, §3.3 Proposition 12]) the sequence
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.
The final strengthening allows one to replace affinoid domains , 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 of we encounter, one shouldn’t be able to produce using another affinoid (not necessarily an affinoid open in and an affinoid cover of 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 admissible if there exists an affinoid (in ) cover (possibly infinite) of such that for any map of affinoid -algebras such that the induced map has one has that the affinoid cover of has a refinement (cf. [Bos, Pg. 83]) by a finite affinoid open cover. Similarly, an admissible covering of the admissible open is a collection of admissible open subsets of which cover and have the property that for any -algebra map of -affinoids such that one has that the cover has a refinement by a finite affinoid open cover.
One should interpret non-admissible covers of an affinoid as ‘bad’ and, in particular, should be disregarded.
Example 6: The cover the open unit disk is an admissible open subset of (e.g. see [Tia, Example 2.1.6]) and is admissible (it’s even rational—it’s ) but the cover 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.
We then have the following general strengthening of Theorem 4 and Theorem 5:
Theorem 7:Let be an affinoid algebra. Then, for any admissible open and any admissible covering of by admissible opens the sequence
Now, we have singled out three (increasingly general) situations where our sheaf theory has worked:
- It works for rational domains in and finite rational covers.
- It works for affinoid domains in 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 and admissible covers (this is the finest replacement -topology [see below] which ‘acts the same’ as the -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 (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 (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 -topology.
Roughly, a topology on a topological space is
- A (usually proper) collection of open subsets of ‘the permissible opens’.
- For each permissible open subset a set of open covers of 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
(where 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 -space . One calls an abelian presheaf on the -space an abelian sheaf if for all and all the presheaf satisfies the sheaf conditions for the cover— in other words the sequence
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 ‘-topology’ is that a -topology on is, essentially, a Grothendieck subtopology of 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])
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 -space and an abelian sheaf on one can define the stalk of at a point . Namely, one can define
where is the set of permissible open subsets of containing .
A locally ringed -space is a -space with a sheaf of rings whose stalks are local rings. If, in addition, is a sheaf of -algebras we call a locally ringed -spaces over . A morphism of locally ringed -spaces and/or locally locally ringed – space over 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 one gets two (major) locally ringed -space over given as the pair
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) -topology on . Unless specified otherwise, the -topology on will always taken to be the strong one. We call a locally ringed -space over of the form (with the strong -topology) an affinoid space over . While is really a pair, we shall, in practice conflate and .
This pair satisfied the expected property:
Lemma 9: Let and be affinoid -algebras. Then, the map
given by the map on global sections is an isomorphism. In other words the functor
is fully faithful.
Proof: See [Bos, §5.3 Corollary 7].
Now, given any locally ringed -space over and any open permissible subset one naturally gets a -topology on (e.g. is the set of elements of contained in ) and one can restrict to get a sheaf on so that the pair is also a locally ringed -space over .
So, with all of this setup we can finally defined rigid analytic varieties a la Tate. Namely, a rigid analytic variety is a pair which is a locally ringed -space over such that there is a permissible open cover of such that is isomorphic to an affinoid space. A morphism of rigid spaces is a morphism of locally ringed -spaces over .
One has a natural generalization of Lemma 9:
Lemma 10: Let and be rigid spaces with affinoid. Then, the natural map
is a bijection.
Proof: See [Bos, §5.3 Corollary 7].
Let us finally note that the set of affinoid open subspaces of a rigid space form a distinguished base for (in the sense of [Vak, §2.5]) and thus if one wants to define a sheaf on it suffices to define a sheaf on the distinguished base of affinoid opens (e.g. by [Vak, Theorem 2.5.1]).
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 is the rigid analytic variety . Let us start to prove some of the basic properties of .
Perhaps the first thing to check is that , as we envisioned it, is quasi-compact and connected. Of course, since is actually a -space (in particular not a topological space) we need to define what this means. But, of course, it’s the obvious thing. Namely a -space is quasi-compact if every permissible cover of has a finite refinement. And, a -space is connected if there does not exist a permissible cover of such that is empty and
To prove that is quasi-compact and connected we make the following general observation:
Proposition 11: Let be an affinoid -algebra. Then, is quasi-compact. Moreover, is connected if and only if has no non-trivial idempotents.
Proof: We tailor build admissible covers so to force the quasi-compactness. Namely, if is an admissible cover of then letting be the identity map (in the definition of admissible) one sees by definition that must admit a finite affinoid refinement.
To see the second claim we proceed as usual. If with admissible then and so has non-trivial idempotents as soon as and are not the zero ring. To see this note that if is a non-empty rigid space then is non-zero. Indeed, let and let be an affinoid neighborhood of . Then, we have a ring map and, in particular, if is the zero ring this forces to be the zero ring which forces to be empty, which is a contradiction.
Conversely, suppose that has non-trivial idempotents so that with not zero rings. Let and . Then I leave it to the reader to check that the non-vanishing locus of and , called and , are open, non-intersecting, and that is an admissible cover of .
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 is an integral domain that 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 -space is irreducible if for all admissible opens we have that . Essentially no rigid space is irreducible.
For example, certainly is not irreducible since, for example,
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 and let . Then,
are disjoint admissible opens. Thus, 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 -topology for a coarser one—the Zariski topology. For this, one can see [Con3].
Functor of points
Let us now contemplate what the functor of points of is. Namely, we have the following observation:
Proposition 13: For any rigid space over is a natural isomorphism of sets
Before we prove this let us define what is. This is a presheaf of -algebras on defined as follows. For an affinoid open we define
where is the ring of power bounded elements of (i.e. those elements such that the set 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, extends uniquely to a sheaf on the -space .
One then sees as the global sections of this sheaf. Intuitively one imagines that is the set of elements of such that (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 and are sheaves for the -topology it suffices to prove this natural isomorphism when is affinoid, say . Note then that by Lemma 9 there is a natural bijection
Now, note that since any -homomorphism is continuous (e.g. see Lemma 38) that such a homomorphism is uniquely determined by where it sends . Namely, if then
Moreover, this also easily shows that is of the form if and only if converges for every sequence in such that . But it’s easy to see that this is equivalent to . The conclusion follows.
In particular, let us note that since is a ring (as we leave to the reader to check) one has that is a ring object in the category of rigid spaces over . This is perhaps not too surprising since is meant to be the rigid geometry avatar of which is itself a ring.
Let us also try and say something slightly more interesting about . Namely, let us define a line bundle on a rigid space as a sheaf of -modules which locally (for the -topology) is isomorphic to (cf. [FdvP, Definition 4.5.1]). As per usual the group of (isomorphism classes of) line bundles on forms a group under tensor product which is called the Picard group of and written . What is the Picard group of ?
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 be the open unit disk thought of as a complex manifold. We claim that . Indeed, we have the exponential sequence
Taking the long exact sequence in cohomology, using the fact that
(e.g. see [Wed, Theorem 11.13]), and the fact that is contractible we get a natural isomorphism
and of course we have natural isomorphisms
Thus, it suffices to show that . This last equality can be shown to be true by the Mittag-Leffler theorem.
So, a guess might be that since the open disk and closed disk are not too far apart that the Picard group of is also trivial. This, as it turns out, is correct:
Proposition 15: Let be an affinoid -algebra. Then, there is a natural isomorphism
given by associating a finitely generated projective -module with the sheaf on which associates to an affinoid open the tensor product .
Proof: See [Bos, §6.1 Theorem 4] or [FvdP, Proposition 4.7.2].
In particular, we see that proving that is equivalent to proving that . But, to do this it suffices (e.g. see [Stacks, 0BCH]) to show that is a PID. But, this is well-known (e.g. see [Bos, §2.2 Corollary 10]).
Defining the dimension for a rigid space 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 is the supremum of as ranges over the points of .
So, what is the dimension of ? Of course the answer is , but we need to verify this. But, as observed above, since is a group under addition it suffices to check . What is this ring? Well, almost by definition (since closed subdisks around are clearly a basis for —cf. our definition of stalk) that
In other words, is nothing but the ring of power series convergent in some neighborhood . This is, by inspection, a DVR. Indeed, it’s easy to see that is the unique maximal ideal of (e.g. it’s the same proof as here) from where one can easily deduce the conclusion.
The punctured closed disk
We now would like to consider the spaces where are any points of . We call such a space a (several times) punctured disk. For sake of simplicity we assume that is algebraically closed (or that the points are -points). This does not change the story in any substantive way, it just allows one to not have to worry about an -point, for finite, splitting in to several points over .
Let us begin with the most basic case: the -analytic variety . There is evidently an automorphism of which takes to . Indeed, this follows from the aforementioned fact that is a ring object over —just take the translation by automorphism of . Thus, we may assume without loss of generality that .
Remark 16: Note that this observation already dispels an easy-to-make misconception about . Namely, if one draws then one has a tendecy to treat points in the ‘boundary’ as being different than points in the interior . But, the above shows that points in really don’t look any different than points in .
For example, a picture of the closed disk lends one to believe that points on the ‘boundary’ don’t have closed disk neighborhoods still contained in the disk. This, of course, is false. If then the set is, in fact, contained in , let alone . This is one of the reasons that notions of ‘boundary’ (e.g. as in Berkovich’s theory of rigid spaces) are more subtle than one might first imagine.
We begin with the following observation:
Proposition 17: The punctured disk is an admissible open subset of . It is connected and not quasi-compact.
Proof: We can essentially kill three birds with one stone. Namely, for each let us set
(this is an example of an annulus). Note that each is an affinoid domain in . In fact, is a rational domain. Indeed, we have that
and evidently generate the unit ideal in since is a unit. The affinoid domain claim then follows from previous comment (cf. [Tia, Example 1.5.12 part 3.]).
Note that is an open cover of because and so if then and thus for some . We claim that is an admissible cover. To do this, let be a morphism such that . Let be the associated map of affinoid algebras and let . Note that since that is a unit in . Indeed, note that we have a factorization of maps of rings
and since is invertible in the invertibility of follows. Let us now consider . Note then that by the maximum modulus principle (e.g. see [BGR, §6.2.1 Proposition 4]) that there exists some such that for all and so for all . Let be large enough so that . Then, it’s clear by this that . So then evidently has a finite refinement as desired.
Now, to see that is connected it suffices to note that is connected. Indeed this would follow from the obvious -topology analogue of the classical statement “if a topological space has an open cover by connected subspaces all of which intersect then is connected”. To see that is connected we can use Proposition 11 and merely note that
and I leave to the reader to check that is a domain.
Finally, to see that is not quasi-compact it suffices to note that has no finite refinement which is easy and left to the reader.
One thing that follows from the above is the following simple observation:
Corollary 18: The -space is not Noetherian.
By Noetherian we mean that not every admissible open subset is quasi-compact. This is, of course, in stark constrast to the algebraic setting where an affine scheme with Noetherian global sections is Noetherian. This is a subtelty that can often lead one astray when applying algebraic geometry intuition to the rigid setting (we’ll see an example of this in another post).
Functor of points
We can now try and understand the functor of points of . But, this has essentially been observed in the process of proving Proposition 17:
Proposition 19: There is a functorial isomorphism
In particular, we see that is a group object in the category of rigid analytic varieties over .
The multi-punctured disk
Let us now turn our attention to the case of a several times punctured closed disk. Namely, let be an integer and consider .
We would first like to give the analogue of Proposition 17 for these multi-punctured spaces:
Proposition 20: The multi-punctured closed disk is an admissible open subset of . It is connected and not quasi-compact.
Proof: To show that is an admissible open subset which is not quasi-compact, one can proceed essentially as in the proof of Lemma 17 (see [BGR, §9.1.4 Proposition 5] for more details).
To prove connectivity one makes the following general topological observation. Let be a connected topological space and let be a closed point. Suppose that has a connected neighborhood of such that is connected, then is connected. Using this, and the fact that every point of has a neighborhood isomorphic to , one reduces the question (inductively) to the case when . The claim then follows from Proposition 17.
Functor of points
We would now like to compute the functor of points of . But, again, this follows exactly as Proposition 19:
Proposition 21: There is a functorial isomorphism
where note that so that the above makes sense.
The Picard group
Finally, we would like to discuss what the Picard group is. To do this, we assume that is actually spherically complete (also called maximally complete)—for a discussion of spherically complete fields see [BGR, §2.4.4]. Then, one can show that . This is substantially harder for than the case discussed in the last section. For a proof one can see [FvdP, Theorem 2.7.6].
Remark 22: I’m actually not sure if the Picard group of the multi-punctured disk is non-trivial if is just assumed algebraically closed and/or complete. See Theorem 25 below for why this is not an unreasonable uncertainty.
Computing the dimension of the multi-punctured disk is easy. Namely, since is a group object we have, as observed before, that for all points we have that
in particular since for any point the local ring agrees with we deduce that
as one would expect.
The open disk
The next example we want to consider is the open disk . More rigorously, we consider the open subset
which we can think of as an open rigid subspace of .
We start our discussion of as in the case of the closed disk/multi-punctured closed disk:
Proposition 23: The open disk is an admissible open subset of which is connected and not quasi-compact.
Proof: As in the proof of Propsition 17 we can kill all three birds with one stone. Namely, one can readily see that
Let us shorten to . Then it’s easy to see that each is an affinoid open subset of (it’s a rational open). We claim that this is an admissible cover, but this follows exactly as in Proposition 17 (it follows from the Maximum Modulus Principle). Moreover, each is connected (use Proposition 11 and the fact that ). Since this implies that is connected. Finally, since for all (e.g. think about -points) we see that we have given an infinite cover of with no finite refinement.
Functor of points
We can, similar to Proposition 13, describe the functor of points of . Namely, we have the following:
Proposition 24: There is a functorial isomorphism
As in Proposition 13 we need to explain our (non-standard) notation here. Namely, we want to define a sheaf on and to do this it suffices, as mentioned before, to define a sheaf on the Grothendieck topology of affinoid opens and affinoid open covers. So, let be an affinoid open. Then, we set
where is the (additive) group of topologically nilpotent elements of (e.g. the set such that —see [Bos, §3.1 Corollary 18] for more details). One can check that this satisfies the sheaf conditions and thus extends to a sheaf on . Of course, we then have that is the group of global sections of .
Proof(of Proposition 24): Since both sides are sheaves for the -topology on it suffices to check functorial agreeance on affinoids. So, let be affinoid open. Then, let us note that we have an injection
where the last identification is by Proposition 13. So then, we need to figure out what subset this is. But, let us note that if corresponds to a map with image in then (in the language of loc. cit.—see also [Bos, §3.1 Proposition 7]) and so . The converse is also clear since if then defines a map . But, note that since that for some (since ). Then, evidently (given Proposition 13 and the decomposition of in Proposition 13) that and thus deos define a map .
The Picard group
The Picard group of is surprisingly, surprisingly subtle. In particular, our Example 14 was a bit of a ruse. Namely, we have the following:
Theorem 25: The following are equivalent:
- is spherically complete.
Proof: See [Gru, V, Proposition 2].
In particular, since is not spherically complete we have that ! Maybe I’ll discuss an example in the future—until then I suggest looking at loc. cit.
Again, for any we have
Affine space (and analytification)
We would now like to describe the rigid space which one can think of as the analytification (a la Serre’s GAGA) of the -scheme . In some sense one can think of it as (e.g. see [Bos, §5.4 Lemma 1] for a discussion in this direction), but we shall not do so.
In general, we define the analytification of a locally of finite type -scheme exactly as in complex geometry. Namely, an analytification of is a rigid space and a map of locally ringed -spaces over
(where has the -toplogy given by its usual Zariski topology) such that for any other rigid analytic variety one has that the natural map
is a bijection (here both sets of maps are of locally -ringed spaces over ).
We then have the usual proposition:
Proposition 26: For every locally of finite type -scheme there exists a unique analytifiation.
Proof: See [Bos, §5.4 Definition/Proposition 3].
The idea is somewhat clear though. Namely, since
one can set
where and we open embed by identifying with . In other words, we think of as the union of increasing closed disks. For an affine finite type -scheme we then set
which, as one can check, is affinoid.
For a general locally of finite type -scheme one covers by finite type affines, shows that the above definition for affine finite type -schemes turns distinguished open embeddings into affinoid open embeddings, then glues. Again, for more details see [Bos, §5.4 Definition/Proposition 3].
We are in this subsection particularly interested in just but it can’t hurt to list some basic properties of the analytification functor
(it is clearly a functor by the universal property of analytification) in general. Namely we have the following:
Proposition 27: Let and be locally of finite type -schemes and let be a map of -schemes.
- There is a canonical bijection of -sets between and .
- is connected if and only if is connected.
- For The natural map is faithfully flat and, moreover, induces an isomorphism on completions.
- is smooth over if and only if is smooth over . More generally, is smooth if and only if is smooth.
- is finite if and only if is finite.
- is reduced if and only if is reduced.
- If is any rigid space and is affine, then the natural map is a bijection.
In the above we used three notions which have not been defined: finite morphisms, smooth morphisms, and reduced rigid analytic varieties.
So, let us say that a morphism of rigid analytic varieties is finite if for all affinoid opens we have that is affinoid open, say equal to , and the induced map is finite.
We will say that is smoothly factorizable if we can factorize as where the first map is étale (this means the usual thing—it’s flat and unramified as a map of locally ringed spaces) and the second map is the projection (cf. the definition of smooth given in [Fu, §2.4]) and we say that is smooth if it’s smoothly factorizable locally on source and target. In particular, is smooth over if is regular local for all .
Finally let us say that a rigid analytic variety is reduced if is reduced for all .
Proof (of Proposition 27): See [Con3, Lemma 5.1.1], [Con3, Lemma 5.1.3], [Con3, Theorem 5.1.3], and [Con3, Theorem 5.2.1].
Of course, one also has a GAGA theorem as in the complex analytic setting:
Theorem 28 (Kopf—rigid analytic GAGA): Let be a projective -scheme. Then, there is an equivalence of categories
such which induces natural isomorphisms
for all .
Proof: The original attribution is [Kop] but an alternative proof, in English, can be seen in [Con2].
In the usual way we deduce the following corollary:
Corollary 29: The analytification functor
is fully faithful.
Of course, we remark that the analytification functor is always faithful, but rarely full for non-projective schemes. For example the natural map
is not surjective since the latter contains something like
which converges everywhere on but is not a polynomial. We will see an even more extreme example of this later on with our discussion of Tate curves.
It is also true that the analytification functor is far from essentially surjective. For example we have the following nice proposition:
Proposition 30: Let be a reduced connected finite type -scheme and an affinoid -algebra. Then, there is no non-constant map .
In particular, the above implies that any reduced and connected rigid space which admits a non-constant map to an affinoid (e.g. an affinoid or the open disk or…) is not the analytification of a -scheme.
Proof (of Proposition 30): I don’t know a canonical reference for this. See [HL, Lemma 5.4] or [JV, Proposition 2.8].
Let us now get in to the discussion of proper. We begin with the following:
Proposition 31: The rigid analytic space is connected and not quasi-compact.
Proof: One could use Lemma 27 part 2. to show that is connected, but that’s too complicated. Namely, one can kill both birds with one stone by observing that, by definition, we have that
and . To see that is connected we merely use Proposition 11. Since no is equal to (by definition) we see that is not quasi-compact.
Functor of points
One can cheat and figure out the functor of points by using Proposition 27 part 8. But, we can prove this directly (and thus prove that really is the analytification):
Proposition 32: There is a functorial identification
Proof: It suffices to check that this equality holds for affinoid since both sides are sheaves for the strong -topology. But, then
where we have used the quasi-compactness of to commute this limit (since the image of must land in some ). But, since we have (exactly as in Proposition 13) an identification
but given the definition of our transition maps we then see that
where the union is taken in . We leave it to the reader to verify that
as desired (hint: show both sides are sheaves and so it suffices to show the result on affinoids and then use the definition in [Tia, Proposition 1.4.13], the fact that , and the fact that is submultaplicative).
The Picard group
Let us now discuss what the Picard group of is. One should be leery about guessing that it’s since, in some sense, is like an ‘infinite radius’ version of the open (poly)disk since it’s an infinite increasing union of closed disks. Thanks to Theorem 25 this might make one guess that perhaps the Picard group is non-trivial. Thankfully this is not the case:
Theorem 33: The group is trivial.
Proof: See [Gru, V, Proposition 2] for a nice discussion of this fact.
Remark 34: Just as an interesting side note it’s actually still an open question as to whether or not the ‘Bass-Quillen conjecture’ is true for —in other words whether all vector bundles are trivial. See for instance see [KST] or [Sig].
The dimension of is what one would expect. Namely, from Proposition 27 part 6. we have that
as desired. Of course, one can also verify this by hand essentially by showing that each is -dimensional. This follows from generalizing our dimension arguments in the case when from above.
We are now interested in studying which, by deifnition for us, is .
We would now like to state the basic topological properties of . Namely, we have the following:
Proposition 34: The space is connected and quasi-compact.
The proof of Proposition 34 can be proven by giving a decomposition of into certain open affinoids. Since this decomposition is independently interesting, we discuss it outside the context of the proof of Proposition 34.
A naive guess is that one can decompose in to open subspace isomorphic to . This is, in fact, true but is not as useful since it’s not a decomposition into affinoids. To decompose in to affinoids we can first remember how the decomposition of into open copies of works. Namely, this decomposition comes by the observation that an element with can be scaled as follows:
One then notes that such a representation is unique and, in fact, the map
(where in the last tuple we omit the entry ) is a bijection.
In the rigid setting we can in fact do better. Namely, let us say that a is unidmodular if for all and for some . Note that by scaling one can assume that every is represented such that is unimodular. One then defines a bijection
(where again we’ve omitted ).
In more rigorously term we have an isomorphism
with (following the nice notation of [Vak, Exercise 4.4.9])
(where again we are omitting ) and we are making the identifications
given by . In fact, one could (perhaps should) take this as our definition of using our description of the functor of points of (using this gluing description as our definition!) to verify it really is an analytification of .
Remark 35: Since we were working at the level of points the above is only really valid over . That said, one can easily check that this decomposition is -invariant and so descends to a decomposition over .
Regardless, this decomposition immediately implies Proposition 34 since each (where we are imprecisely aslo denoting by image under the quotient map ) is connected and quasi-compact, and for all .
Functor of points
The functor of points of is what one expects from its corresponding description in algebraic geometry. Namely:
Theorem 36: There is a functorial bijection
Here we say, as in algebraic geometry, a set of is globally generating if for all in one has that the image of under the map
is a generating set of the -module . Moreover, we say that
if there exists an isomorphism carrying to .
The proof of Proposition 36 is exactly in the scheme theoretic situation, and we leave it to the reader to check this.
One can use Rigid Analytic GAGA to see that
and, moreover, that a generator for is given by . Concretely one can think about this as the line bundle obtained on
which is given by the trivial line bundle on each and which is glued via the overlap isomorphisms .
If one doesn’t want to use the full power of Rigid Analytic GAGA one might try to use the isomorphism
(which is true for all locally -ringed space—e.g. see [Stacks, Tag040E] and [Stacks, Tag0A6G]) together with the non-trivial fact that constitutes a Leray cover (i.e. a covering as in [Stacks,03F7]) to deduce that
from which one really can deduce that is a generator by the usual computation. The fact that really is a Leray cover, the fact that has vanishing higher coherent cohomology, is merely a souping up of Theorem 7 (e.g. see [BGR, §8.2.1 Theorem 1]).
Given our decomposition
we know that
as one would expect.
The multiplicative group
We now move on the multiplicative group which, by definition for us, is the analytification of the multiplicative group .
We start with the following topological description of :
Theorem 37: The topological space is connected but not quasi-compact.
Again, we might try to apply Proposition 27 part 2., but this is wholly overkill. In fact, both of these properties follow from the following generally useful decomposition of . Namely, recall that we have the decomposition
with each . From this, we see that we have the decomposition
where one can check that . Since each intersects all the other , and each are connected (by Proposition 17) we deduce that is connected. It’s clear that no finite subset of covers and thus we also deduce that is not quasi-compact.
Functor of points
We now move on to the description of the functor of points of . Of course, we expect that it spits out the units of the global sections of a rigid analytic variety and this actually follows from Proposition 27 part 8. But, again, we can see this in a much more down-to-earth fashion:
Theorem 38: There is a functorial bijection
Proof: The proof of this is essentially exactly as in Proposition 32. Namely, we see that it suffices to check this for an affinoid space . From there we use the quasi-compactness of to write
to which one then applies Propositition 19 and the decomposition
The Picard group
The Picard group of is again slightly more subtle than one might bargain for. That said, we do have the following result:
Theorem 39: If is spherically complete then .
Proof: This follows from [FvdP, Theorem 2.7.6].
It’s likely that this theorem holds without the assumption that is sphereically complete, but I’m not sure.
The dimension of is as one would expect. This follows since each has dimension by our previous discussion about the dimension of multi-punctured disks.
Tate elliptic curves
So, let us consider the meromorphic function field
where more explicitly one can write
which is a domain as one can easily check. Let us now fix in with . We can then consider the subset
of which is, in fact, is a subfield as one can easily check.
The realization that Tate had, which was one of the first motivating reasons to want to create a theory of rigid geometry, was the following beautiful result:
Theorem 40 (Tate): There exists a unique elliptic curve over such that . Moreover, one has that
Moreover, one has that there is a functorial isomorphism
for every extension of . Moreover, the map
is a bijection with analytic inverse.
Proof: See [Lut, Theorem 1.1] and [Lut, Theorem 1.2] and the references therein.
Now, one is tempted to think that the isomorphism
could be upgraded to an isomorphism
in some sense. Of course, we have to make sense of what this quotient means on the right-hand side.
To do this, we need to give a decomposition of which is shuffled around by the action of . Namely, let us define the following affinoid domains inside of
One note that
Both of these spaces are annuli and they have an outer circle and an inner circle defined as further rational domains inside of . We denote the outer circles of by and the inner circles by . Evidently we have that .
Note that for each we have a natural multiplication-by--map
which for an affinoid algebra is just the map
This is an isomorphism of rigid analytic varieties. We note then that one has . Moreover
gives an open cover of .
From this, it seems reasonable to define to be
which which we mean the rigid space obtained by gluing to itself along the isomorphism (gluings along open subspaces clearly exist in the category of rigid analytic spaces).
We then have the following justification for writing this as a quotient
Proposition 41: The obvious map
is -invariant for all and is universal for such.
The beautiful tie-in with Theorem 40, as we’d hoped, is the following:
Theorem 42(Tate): There is an isomorphism of rigid analytic varieties .
Proof: See [Lut, Theorem 1.2] and the references therein.
We call rigid analytic varieties of the form Tate elliptic curves.
Note that we really needed to be working in the rigid category to have an isomorphism
for two distinct, but equally important, reasons:
- The map has infinite fibers. Note that any infinite subset of is dense, and thus one cannot even have a non-constant map from with infinite fibers.
- We get from Theorem 41, in particular, a surjective map . Note that any algebraic map (for any elliptic curve over ) is constant since otherwise it would give rise to a surjective map which is impossible by the Riemann–Hurwitz formula (e.g. see [Har, Example 2.5.4]).
Remark 43: As mentioned earlier we see from 2. the striking non-fullness of the map
since the former set is consists only of constant maps but the latter set contains a topological cover!.
The topological properties of Tate elliptic curves are quite simple. Namely, they are quasi-compact and connected. The quasi-compactness comes from the fact that they are a quotient of the compact space
and they are connected since the images of each copy of in are connected and intersect non-trivial.
Functor of points
The functor of points of is not particularly easy to describe since it is, after all, related to the functor of points of which has no explicit description.
The Picard group
By Theorem 28 we know that we have an isomorphism
and thus it suffices to understand the Picard group of . But,
where are the degree line bundles in on . Moreover, we know that
where the first isomorphism has inverse
where is the identity element, and the second isomorphism is from Theorem 40. To understand the composition explicitly, if one fixes in one gets a line bundle on by gluing the trivial line bundles on via the transition map where acts as multiplication by on the trivial bundle. This does descend to a line bundle on . Moreover, this line bundle only depends on up to its class in .
The dimension of is obviously one. Indeed, has dimension , and since is obtained by gluing to itself along an open subset, the conclusion follows.
Since one of our main motivations for defining rigid analytic geometry is to study arithmetic geometry we would be remiss to not give some number theoretic applications of the theory of Tate elliptic curves.
Reduction type and Tate’s theorem
To explain these results, it is first helpful refine the results of Theorem 40 and Theorem 42 slightly. To do this, let us first recall some basic theory of elliptic curves over a -adic field . We shall denote the ring of integers of by and the residue field of by (don’t confuse this with our analytic field from above!). We shall denote the valuation on (the unique one extending the -adic valuation on ) by and we shal ldenote the associated absolute value on by .
For the sake of simplicity, we assume that . This will allow us to work with simplified Weierstrass models.
We then recall, following [Con4], that a Weierstrass model of is a relative cubic curve over
together with an isomorphism . Again, note that [Con4] requires that is actually an essentially general cubic, but since we have that is invertible in and so we can always put our elliptic curve in this form (e.g. see [KM, (2.2.6)]). We define the discriminant of , denoted , as follows:
One can check that is indpendent of the isomorphism class of . One can give a more highfalutin description of the discriminant as in [Con4] where this becomes more obvious.
We then note that is smooth, and thus an elliptic scheme (in the sense of [KM, (2.1.1)]) if and only if . In general there won’t exist a Weierstrass model of which is smooth, and the best thing we can ask is that is a minimal Weierstrass model which means that is minimal amongst Weierstrass models for . A priori if one has two minimal Weierstrass models and one only knows that and, in particular, there is no reason for to be isomorphic to . This is the case though. This can be seen by explicit calculation (as in [Sil, 1.3 in VII]) or by abstract theory (as in [Con4, Corollary 4.6]).
Regardless, we now know that a minimal Weierstrass model for exists and is unique up to isomorphism. We then define the reduction of to be the curve .
Of course, this reduction is a priori just just a cubic curve embeddable in . In particular, there is no reason that should be smooth over . That said, is necessarily very mildly singular (if it’s singular at all):
Proposition 44: Let be any field and let be a singular cubic hypersurface in . Then, has exactly one singular point and is in .
Proof: Suppose first that is algebraically closed. Let and be points of which are both singular. Note then that there exists a line such and both lie in . Note then that by Bézout’s theorem (e.g. see [Liu, Corollary 1.20 in §9.1.2]) we have that the and in particular we see that one of or (using [Lieu, Definition 1.1 §9.1.1]) must be . Without loss of generality we may assume that . But, this implies that is smooth at —intuitively this means that and meet transversally at so that
(where is the tangent space) which then implies that has dimension , which means that is smooth at (for details see [Liu, Proposition 1.8 in §9.1.1]). This is a contradiction.
Suppose now that is not algebraically closed and assume that is a singular point of . Note that if is not a -point of then the preimage of under the map has more than point. Each of these points is necessarily singular which contradicts the first paragraph.
We thus can try to understand the geometric properties of , and thus the properties of , by trying to suss out the properties of the singularity of or, equivalently (perhaps not so obviously) the properties of of the smooth locus which, by Proposition 44, is just for some -point in . Let us note that, as can be checked by hand, the point in is smooth and so, in particular, .
Before we state the marquee classification in this direction, we first note the following beautiful fact:
Proposition 45: There exists the unique structure of a group variety on such that is the identity element. Moreover, is smooth, integral, and dimension . If is not smooth then is affine.
Proof(Sketch): The point is that the chord-tangent construction as in the theory of elliptic curves works here as well.
More explicitly, let be distinct. There exists a unique line such that passes through and . If we set to be the unique line in passing through and which is tangent to at .
Note then that by Bézout’s theorem that for any (possibly with ) we have that is a finite -scheme with -dimensional global sections. If then it’s easy to see that is the disjoint union of three -points of which are and a third point . Moreover, note that necessarily is another point of since the intersection of and neccessarily has multiplicity (cf. see the proof of Proposition 44). If then is the disjoint union of (where for a closed subscheme of we denote by the ideal of induced by ) and an -point . Again, since the multiplicity of at is , we see that is in . In either case we see that we obtain a third point from .
Note then the exact same ideas yield associated to the pair a line and a third point of intersection of the line and which is in . We denote this third -point of by .
The proof that this process gives a group structure on is the same as in [Sil, Proposition 2.2 in §III.2] and the fact that this is algebraic and defined over is the same as in [Sil, Group Law Algorithm 2.3 in §III.2].
Let us now verify the claimed geometric properties of . The fact that it’s smooth and dimension are obvious, and thus it suffices to prove that it’s integral. Since it’s smooth it suffices to prove it’s connected (e.g. by [Stacks, Tag033M]). But, to do this it suffices to show that is irreducible. But, this is clear since
where and are the images of and in . In particular, note that the equation
is evidently irreducible by setting and arguing that any factorization of would have a non-zero term. From this, the claim that is irreducible is clear from basic theory.
Finally we verify the affineness claim. But, this follows from totally general theory (e.g. see Lemma 13 from this post).
Thus, a natural question is what group is. As it turns out there are not many options:
Proposition 46: Let be a finite field and a smooth integral connected group variety of dimension . Then, either is an elliptic curve or it’s isomorphic to one of the following three groups:
- The additive group .
- The split multiplicative group .
- The non-split multiplicative group .
Proof: This follows from Theorem 29 of this post.
Here we using the notation to denote the norm elements of the Weil restriction (e.g. see [Mil1, §2.i]).
Abstractly this is the group scheme which associates to an -algebra the group
where is defined to be the determinant of the -linear map
which makes sense since is a free -module of rank .
Concretely if for then
can be thought about as the subgroup of given as the set of matrices of the form
with the usual matrix multiplication.
So, with this setup we can divide up the elliptic curves over into categories depending on which group is which constitutes four cases:
- If is an elliptic curve we say that has good reduction.
- If is the additive group we say that has additive reduction.
- If is the split multiplicative group we say that has split multiplicative reduction.
- If is the non-split multiplicative group we say that has non-split multiplicative reduction.
There is further terminology concerning reduction type that is useful. Namely, if has either split or non-split multiplicative reduction, we say that has multiplicative reduction. If has either good reduction or multiplicative reduction then is said to have semistable reduction. If has either good or additive reduction it is said to have potentially good reduction. The reason for this naming scheme will soon be clear.
As we said before, we can also understand the reduction type of in terms of the singularity of at its singular point (if it has one) as follows:
Proposition 47: Let be an elliptic curve over and assume that . Then, the following are equivalent:
- has good reduction.
- If one writes as the projective closure of , then is separable.
So are equivalent the following:
- has additive reduction.
- The completed local ring is isomorphic as -algebra to .
- and .
- If one writes as the projective closure of , then has a repeated root of multiplicity .
So are equivalent the following:
- has split multiplicative reduction.
- The completed local ring is as a -algebra isomorphic to .
- , , and is a square in .
- If one writes as the projective closure of , then can be factorized in in the form for some .
So are equivalent the following:
- has non-split multiplicative reduction.
- The completed local ring is isomorphic as a -algebra to where is an irreducible homogenous polynomial of degree .
- , , and is not a square in .
- If one writes as the projective closure of , then is irreducible in .
Proof: All of these are fairly straightforward and are covered in [Sil, Proposition 5.1 in §VII.5] except the statements concerning the completed local rings, but these can be checked elementarily given the explicit description of the polynomials cutting out in as explained in each part.
Using these characterizations, especially those involving the valuation of , it’s not super hard to deduce the following:
Corollary 48 (Semistable reduction theorem): Let be an elliptic curve over . Then, if has multiplicative reduction then so does for every finite extension . If has good reduction, then so does for every finite extension . If has additive reduction, then there exists a finite extension such that has good reduction.
Proof: See [Sil, Proposition 5.4 in §VII.5]. .
Remark 49: Be careful to not make the following mistake. It seems tempting to think the above theorem is wrong, for examples that additive reduction becomes good reduction over an extension, given Proposition 47. Namely, if one assumes that the minimal Weierstrass model of is . That said, this claim only necessarily holds true if is unramified (i.e. that is étale). So, the extension which changes an additive reduction curve to good reduction is necessarily ramified.
Another useful thing that follows from Proposition 47 is the following:
Corollary 50: An elliptic curve over has multiplicative reduction if and only if . Consequently, has potentially good reduction if and only if .
Proof: For example, see [Sil, Proposition 5.5 in §VII.5].
We can then understand Tate’s theorem as follows:
Theorem (Tate) 51: The map
given by is a bijection.
Proof: See, for example, [Sil1, Theorem 5.3 in §V.5].
This makes intuitive sense since one might imagine that should have the result of since projects to in , and this looks quite a bit like should have split multiplicative reduction. Of course, this is clearly missing part of the picture since, for example, we don’t see where the singularity of is entering into the picture. We will revisit this discussion when we discuss Raynaud’s perspective of rigid geometry.
Now that we have a more refined sense of which elliptic curves are Tate elliptic curves, they are precisely the ones with split multiplicative reduction, we can now more properly state the arithmetic applications of the theory.
The first application is to the study of the -adic Tate module of an elliptic curve over . Namely, recall that if is an elliptic curve then its -adic Tate module is the representation given by
where is the closed subgroup scheme of given by the -torsion and the transition maps are the multiplication-by- maps (e.g. see [Sai, §1.3] or [Sil, §III.7]). We also have the rational -adic Tate module which is the -representation given by
Note that as an abstract -module one has that is isomorphic to and thus is isomorphic abstractly to as a -vector space (e.g. see [Sai, Proposition 1.19] or [Sil, Proposition 7.1 in §III.7]).
Remark 52: More abstract one has identifications of Galois representations
where these objects are the étale cohomology groups of , but we won’t really use or need this fact. See this post for more details.
Using the theory of Tate elliptic curves we deduce the following:
Proposition 53: Let be an elliptic curve over with split multiplicative reduction. Then, for every one has an extension of Galois representations
which is non-split for sufficiently large and therefore one has a non-split extensions
Here we are using the notation to denote the -representation given by and by the representation with the transition maps being the -power maps. We are also using and to denote the trivial Galois representation on these groups.
Proof (of Proposition 53): Let us make an identification for some . Let us also fix for each a primitive root of unity in and a -th root of . Let’s assume that we’ve made this choice compatabily so that
for all .
Let us then note that from the isomorphism of
we easily see that
as Galois representations. But, evidently we then have the subrepresentation given by and generated by . Note moreover that for each in one has that we can write for a unique . Thus, we see that the quotient is isomorphic to with the trivial action as desired.
The fact that this sequence doesn’t split for sufficiently large is clear. Indeed, such a section for the projection map must send to for some and for this section to be Galois equivariant we need that this element is Galois fixed. This implies that is in , and evidently if is sufficiently large this is impossible.
The -adic statement follows by passing to the limit. Checking non-splitting at the rational level requires a small argument, but we leave this to the reader.
In fact, we can elaborate on this result even further. Namely, note that one has
where the first object is the set of isomorphism classes of extensions in the category of Galois -representations, and the second equality follows from Kummer theory (e.g. see this post). We can then understand the above as saying that the association of an elliptic curve with split multiplicative reduction to the extension class of in is the map sending to the class of in . The non-splitness is then clear since is not in for sufficiently large . The non-splitting of the rational Tate module is then clear since the natural map
Of course, one might be interested in understanding what happens in the case of non-split multiplicative reduction. But, this is quite simple as the following shows:
Proposition 54: Let and over any field and suppose that becomes isomorphic to upon base extension to the quadratic extension . Then, there is an isomorphism of Galois representations
where is the quadratic character associated to .
Proof: For example this follows from [Sil1, Lemma 5.2(c) in §V.5].
Here the quadratic character associated to is the unique non-trivial homomorphism factoring through . By we mean the twist of by which means the same underlying module of but with the twisted Galois action. Namely, for and if denotes the action of on in , then the action of on in is .
Note that one interesting consequence of the above discussion is that -adic Tate modules (and thus -adic étale cohomology groups by Remark 52) needn’t be semisimple Galois representations.
One of the first major theorems that one proves about the arithmetic of elliptic curves is the Mordell–Weil theorem which states that if is an elliptic curve over a number field , then is finitely generate. This result requires quite a bit of sophistication in the form of the weak Mordell–Weil theorem (e.g. see this post) and techniques using height functions (see [Sil, §VIII.4-§VIII.6] for details).
One consequence of this is that the group of torsion points of is finite. That said, one can get a significantly easier proof of this, at least assuming the Tate theory, of elliptic curves. The point of this is not necessarily to claim that this is a ‘more elementary’ or ‘simpler’ proof than the usual proof of the Mordell–Weil theorem (or, of course, this can be deduced more simply by using Nagell–Lutz and some theory about reduction—for example see [Mil2, §II.4]) but to illustrate that we can get a purely arithmetic-geometric result by analytic methods.
Regardless, we see the following result quite easily:
Proposition 55: Let be an elliptic curve over a number field . Suppose that has a prime such that has multiplicative reduction. Then, is finite.
Proof: Set to either be , if has split multiplicative reduction over , or the quadratic extension of that splits if has non-split multiplicative reduction. Note then that we have an injection
and we know from Tate’s theorem that
for some with . That said, is clasically isomorphic as a group to
where is the set of roots of unity in (which is finite) and is a finite free -module (e.g. see [Lor, §25.4]). But, this group has the property that all of its quotients have finite torsion from where the claim follows.
Of course, by Corollary 50 we know that has
As a last example, we borrow the following result from [Sil1, §V.6] for which a -adic proof using the theory of Tate elliptic curves is due to Serre.
Namely, let us recall that if is a characteristic field and is an elliptic curve over then we denote by the ring of endomorphisms of -group scheme . As it turns out, there are not many options for the ring . Namely, this ring must either be or an order in an imaginary quadratic extension of (e.g. see [Sil, Corollary 9.4 in §III.9]). We say that has geometric complex multiplication if is an order in a quadratic imaginary extension of
Despite first appearances, curves with geometric complex multiplication play a profound role not only in the theory of elliptic curves. Namely, they are certain ‘special’ (or ‘exceptional’) types of curves that are often exceptions for many of the deep theorems in the arithmetic of elliptic curves (cf. [Sil, Theorem 4.9 in §V.4] or [Sil, Corollary 6.3 in §IV.6]) but also serve as the conceptual basis for the theory of canonical models of Shimura varieties (e.g. see [Mil3, §10]).
For this reason, any statement which helps understand elliptic curves with geometric complex multiplication are incredibly significant from the point of view of arithmetic. It is then interesting that Serre was able to prove the following theorem using the theory of -adic Tate curves:
Proposition 56: Let be a number field and an elliptic curve over . If has geometric complex multiplication then .
Proof: For a proof using the theory of Tate elliptic curves see [Sil1, §V.6].
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