Rigid analytic spaces
Gregor Bruns 04.05.2016
Most of the material in this talk is stolen from [Bos14]. I claim no originality.
1 Overview
Throughout these notes letKbe a complete non-Archimedean field. Here we will develop the notion of a classical rigid spaceXoverK. It will consist of a set of points with values in an algebraic closure ofK, together with a notion of analytic functions on subsets. These functions arise by piecing together elements of affinoidK-algebras.
The spaceXcarries a canonical topology induced by the topology ofK. As a conse- quence of Tate’s Acyclicity Theorem 3.2 however, we cannot admit all open subsets ofX to construct a structure sheaf of analytic functions. Instead we have to restrict ourselves toadmissible open subsetsand theiradmissible open coverings. Sheaves onXwill be considered relative to this additional structure of a Grothendieck topology.
This entails, for instance, that there are non-zero abelian sheaves on a rigid space with all stalks being zero. Hence there seem to be not enough points in rigid analytic spaces.
The idea ofBerkovich spacesis to consider additional points with values in fieldsL(not necessarily finite overK) together with a choice of a non-ArchimedeanR-valued absolute value extending the one onK. Although the construction in general is less natural than one would like, one obtains a Hausdorff topology and in general Berkovich spaces have many nice properties.
The notion of a classical rigid space can also be extended by considering rigid geometry in terms of formal schemes.
2 Affinoid spaces and affinoid subdomains
2.1 Affinoid spaces and the Zariski topology
LetAbe an affinoidK-algebra. Everyf∈Acan be viewed as a function on the set of maximal ideals ofAin the following way. Ifxis a maximal ideal ofAandA/xis the residue field thenf(x)is defined to be the residue class offinA/x. If we embedKinto an algebraic closureK, thenf(x)is not defined uniquely (only up to conjugation overK),
but the absolute value|f(x)|is. We set|f|sup =supx∈MaxA|f(x)|where MaxAis the set of maximal ideals ofA.
Definition 2.1. LetAbe an affinoidK-algebra. TheaffinoidK-spaceSp(A)associated toAis the set of maximal ideals ofA(aka the maximal spectrum ofA) together with the K-algebraAof functions on it.
For now we have to restrict ourselves to maximal ideals since taking the whole spectrum ofAleads to a lot of unwanted behaviour. For instance, there may be open subsetsU⊆X and closed subsetsY⊆Usuch that after taking the closureYinXwe getY(Y∩U.
Definition 2.2. Amorphismσ: Sp(A) → Sp(B)of affinoidK-spacesis induced by a morphismσ∗:B→Aof affinoidK-algebras by setting
σ(m) = (σ∗)−1(m)
Note thatσ(m)is maximal since we haveK,→B/σ∗−1(m),→A/m.
Now we can define theZariski topologyon Sp(A)as one would expect. The closed sets are given by
V(a) ={x∈Sp(A)|f(x) =0 for allf∈a}={x∈Sp(A)|a⊆x}
wherea⊆Ais any ideal. We then get the usual basisDf of Zariski open subsets and we get an analogue of Hilbert’s Nullstellensatz. We also can make sense of fiber products of affinoidK-spaces using completed tensor products.
2.2 The canonical topology and affinoid subdomains
LetX=Sp(A)be an affinoidK-space. Since the Zariski topology is very coarse, we are going to introduce another topology that is induced from the topology onK(which we already studied a bit). In particular it will share the strange feature of being totally disconnected. We consider sets of the form
X(f,ε) ={x∈X:|f(x)|6ε} wheref∈Aandε∈R+. They will be our opens.
Definition 2.3. Thecanonical topologyofXis the one generated by allX(f,ε).
We writeX(f) =X(f, 1)and
X(f1, . . . ,fr) =X(f1)∩ · · ·X(fr) ={x∈X:|fi(x)|61 for alli=1, . . . ,r} forf1, . . . ,fr∈A. These sets are calledWeierstraß domainsinX.
Lemma 2.4. The canonical topology onXis generated by the subsetsX(f)wheref∈A.
In particular, a subsetU⊆Xis open if and only if it is a union of Weierstraß domains.
As an exercise in how to work with these things, here is a useful lemma that we can employ to prove the openness of various other sets.
Lemma 2.5. Letf∈Aandx∈Sp(A)such thatε=|f(x)|>0. Then there is ag∈Awith g(x) =0 such that|f(y)|=εfor ally∈X(g). In particular,X(g)is an open neighborhood ofxcontained in{y∈X:|f(y)|=ε}.
Proof. Writef(x)for the residue class offinA/x and letP(ζ) ∈ K[ζ]be the minimal polynomial off(x)overK. FactorP(ζ) =Qn
i=1(ζ−αi)inKand choose an embedding A/x,→K. Then we have
ε=|f(x)| =|αi|
for allisince all roots of the minimal polynomial are conjugate to each other, and the valuation is unique and does not depend on the embedding ofA/xintoK.
Now considerh=P(f)∈A. Thenh(x) =P(f(x)) =0. Lety∈Xand assume|f(y)|6=ε.
Choose an embeddingA/y,→K. Then
|f(y) −αi|=max(|f(y)|,|αi|)>|αi|=ε
for alli(we have equality in the non-Archimedean triangle inequality if both arguments do not have the same value), and therefore
|h(y)|=|P(f(y))|= Yn i=1
|f(y) −αi|>εn
Hence if|h(y)| < εnthen|f(y)| = ε. Taking anyc∈ K∗with|c| < εnwe setg = c−1h
and obtain that|f(y)|=εfor anyy∈X(g).
Corollary 2.6. For f ∈ Aand ε ∈ R+ the following sets are open in the canonical topology:
{x∈Sp(A) :f(x)6=0}
{x∈Sp(A) :|f(x)|6ε} {x∈Sp(A) :|f(x)|=ε} {x∈Sp(A) :|f(x)|>ε}
Proposition 2.7. Letϕ: Sp(B)→Sp(A)be a morphism of affinoidK-spaces and letf1, . . . ,fr∈ A. Then we have
ϕ−1(Sp(A)(f1, . . . ,fr)) =Sp(B)(ϕ∗(f1), . . . ,ϕ∗(fr)) In particular,ϕis continuous with respect to the canonical topology.
Apart from Weierstraß domains there are some more general distinguished subsets of affinoidK-spaces. We call subsets of the form
X(f1, . . . ,fr,g−11 , . . . ,g−1s ) =
x∈X:|fi(x)|61,|gj(x)|>1
Laurent domains. Subsets of type X
f1
f0, . . . ,fr f0
=
x∈X:|fi(x)|6|f0(x)|
forf0, . . . ,fr∈Awithout common zeros are calledrational domains. The openness of these domains follows from our Lemma 2.5.
Weierstraß, Laurent and rational domains are examples of affinoid subdomains ofX. These form the class of subsets that will be ultimately relevant to us.
Definition 2.8. A subset U ⊆ X is called anaffinoid subdomainofX if there exists a morphism of affinoidK-spaces ι:X0 → X such that ι(X0) ⊆ Uand the following universal property holds: Any morphism of affinoid K-spaces ϕ: Y → Xsatisfying ϕ(Y)⊆Uadmits a unique factorization throughι:X0→Xvia a morphism of affinoid K-spacesϕ0:Y→X0.
This definition might seem strange at first since there is no mention ofUbeing open.
As a first indication that it is not so bad, we have:
Example 2.9. Points {x} ⊆ Xare not affinoid subdomains. The problem here is that the universal property does not hold. For simplicity assumex∈Xcorresponds to the maximal idealmx ⊆Awith residue fieldKand{x}as an affinoidK-space comes from the algebraA/mx. Then the inclusion{x},→Xcorresponds to the morphismA→A/mx=K.
But we can also construct the mapϕ:A→A/m2x. On the level of points this corresponds to a map with image{x}. But there is no wayϕfactors overA→K.
If we want to be more rigorous, we have to exclude all possible affinoid K-space structures on{x}. We then need to show that the only ring which could have the universal property is the localizationAmx, but this is not finitely generated and hence notK-affinoid.
By applying the universal property for the inclusions of each pointx∈UinU, we can actually show thatιis injective andι(X0) =U. Hence we can identify the setUwith the set ofX0and get an induced structure ofK-affinoid space onU. We will later see that any affinoid subdomain ofXis indeed open with respect to the canonical topology.
Proposition 2.10. Weierstraß, Laurent and rational domains in an affinoidK-spaceXare open affinoid subdomains.
Proof. We only prove this for a Weierstraß domainX(f)⊆X, wherefstands for a tuple f1, . . . ,fr ∈A. Consider the affinoidK-algebraAhζ1, . . . ,ζriof restricted power series and consider
Ahfi=Ahf1, . . . ,fri=Ahζ1, . . . ,ζri/(ζi−fi)
as an affinoidK-algebra. We have a canonical morphismι∗:A → Ahfiof affinoidK- algebras and an associated morphismι: Sp(Ahfi) → Xof affinoidK-spaces. We will show that im(ι)⊆X(f)andιsatisfies the universal property.
Consider a morphism ϕ: Y → X of affinoidK-spaces with associated morphism ϕ∗:A→Bof affinoidK-algebras. For anyy∈Ywe get
|ϕ∗(fi)(y)|=|fi(ϕ(y))|
since we have an inclusionA/ϕ(y),→B/yof finite extensions ofK, induced byϕ∗. Then ϕ(Y)⊆X(f)if|ϕ∗(fi)|sup 61 for alli=1, . . . ,r. Now considerϕ=ι. Hereι∗(fi)is the residue class ofζiinAhfi, hence its Gauß norm is 1. Since the supremum norm ofι∗(fi) is bounded by the residue norm, which in turn is bounded by the Gauß norm ofι∗(fi), we have shownι(X0)⊆X(f).
We still have to show the universal property forι∗, i.e. each morphismϕ∗:A→ B of affinoidK-algebras satisfying |ϕ∗(fi)|sup 6 1 for all i = 1, . . . ,r admits a unique factorization throughι∗. We can extendϕ∗ to a morphism Ahζi → Bby mapping ζi 7→ ϕ∗(fi). Then allζi−fibelong to the kernel and we get an induced morphism Ahfi →Bwhich is seen to be the required factorization. Uniqueness follows from the
fact that the image ofAis dense inAhfi.
We call these open affinoid subdomainsspecial. Here are some results about properties of (special) affinoid subdomains that one would expect or hope to have.
Proposition 2.11. Letϕ:Y → Xbe a morphism of affinoidK-spaces and letX0 ,→ Xbe an affinoid subdomain. ThenY0=ϕ−1(X0)is an affinoid subdomain ofYand we haveY0=Y×XX0. IfX0is a Weierstraß, Laurent or rational subdomain ofXthen the corresponding statement is true forY0as an affinoid subdomain ofY. The defining functions ofX0pull back to the defining functions ofY0.
Idea of proof. The first assertion follows from the universal property of affinoid subdo- mains, together with the fact that fibered products exist for affinoidK-spaces. The second
part is mainly an application of Proposition 2.7.
Proposition 2.12. Let U,V ⊆ Xbe affinoid subdomains of X. Then U∩V is an affinoid subdomain ofXas well. Furthermore, ifUandVare Weierstraß (resp. Laurent, resp. rational) subdomains then the same is true forU∩V.
Proposition 2.13. LetU → Xbe a morphism of affinoidK-spaces definingUas an affinoid subdomain ofX. ThenUis open inXand the canonical topology ofXrestricts to the one ofU.
Theorem 2.14 (Gerritzen–Grauert). Let X be an affinoidK-space andU ⊆ Xan affinoid subdomain. ThenUis a finite union of rational subdomains ofX.
3 Tate’s Acyclity Theorem
We letXbe an affinoidK-space andTbe the category of affinoid subdomains inX, where the morphisms are the inclusions. Then it makes sense to consider (pre)sheaves onTand we can in particular ask whether the presheafOXof affinoid functions onXis a sheaf.
The first sheaf condition, that locally zero functions are globally zero, is satisfied for OX. On the other hand, because the canonical topology onXis totally disconnected, we usually cannot glue local sections together. However we will see that the gluing condition is satisfied forfinitecoverings.
LetFbe a presheaf onXandU = (Ui)i∈Ibe a covering ofXby affinoid subdomains Ui. We say thatFis aU-sheafif for all affinoid subdomainsU⊆Xthe sequence
OX(U)→Y
i∈I
OX(Ui∩U)⇒ Y
i,j∈I
OX(Ui∩Uj∩U)
is exact.
Theorem 3.1(Tate). LetXbe an affinoidK-space. The presheafOXof affinoid functions is a U-sheaf onXfor all finite coveringsU = (Ui)i∈IofXby affinoid subdomainsUi⊆X.
Idea of proof. One first reduces to Laurent domains, then to Laurent domains generated by one function, and proves it for these domains by direct computation.
We say that a coveringUisF-acyclicifFsatisfies the sheaf properties forUand the Čech cohomology groupsHq(U,F)vanish forq >0.
Theorem 3.2(Tate’s Acyclity Theorem). LetXbe an affinoidK-space andUbe a finite covering ofXby affinoid subdomains. ThenUis acyclic with respect to the presheafOX.
4 Grothendieck topologies on affinoid spaces
We have seen thatOXon an affinoidK-space carrying the canonical topology will usually not be a sheaf. Instead of further improving our spaces, we will rethink our definition of sheaf, or more precisely our notion of open cover. As a rough idea, we are going to adapt the notion of cover so as to only include finite unions of affinoid subdomains. We have already seen in the last section thatOXis a sheaf with respect to these sort of covers. In this way we arrive at a more restrictive notion of analytic function than our naive idea in the beginning: they can only be pieced together from finitely many locally analytic functions on affinoid subdomains.
To formalize our ideas, we make a series of definitions.
Definition 4.1(Grothendieck topology). AGrothendieck topologyTis a category CatT together with a set CovTof families(Ui→U)i∈Iof morphisms in CatT, which we call thecoverings, such that:
1. Ifφ:U→Vis an isomorphism in CatTthen(φ)∈CovT.
2. If(Ui→U)i∈Iand(Vij→Ui)j∈Ji are coverings, then(Vij→Ui→U)i∈I,j∈Jiis a covering.
3. If (Ui → U)i∈I is a covering and V → Ua morphism in CatTthen the fiber productsUi×UVexist in CatTand(Ui×UV→V)i∈Iis a covering.
Grothendieck topologies are not topologies in the usual sense. Instead of capturing the properties of open sets, they only know about what it means to be an (open) cover.
Example 4.2. LetXbe a topological space. Define a Grothendieck topologyTby letting CatTbe the usual category Top(X), i.e. the objects are the open sets and all the morphisms are given by inclusions of open sets. Furthermore, let CovTbe given by the usual open covers, i.e. a family (Ui)i∈I of open subsets of U is an open cover ofU if the map S
i∈IUi→Uinduced by the inclusions is surjective.
Observe that forU1,U2⊆Uopen sets we haveU1×UU2 =U1∩U2. Then the second condition in the definition of Grothendieck topology means that covers can be refined and the third condition means that given an inclusion of open setsV ,→Uand(Ui)i∈Ia covering ofU, the family(Ui∩V)i∈Iis a covering ofV.
It turns out that Grothendieck topologies provide exactly the framework we need to define (pre)sheaves.
Definition 4.3. LetTbe a Grothendieck topology andCa category where products exist.
ApresheafonTwith values inCis a contravariant functorF: CatT→C. We say thatF is asheafif
F(U)→Y
i∈I
F(Ui)⇒ Y
i,j∈I
F(Ui×UUj)
is exact for all coverings(Ui→U)i∈Iin CovT.
Remark 4.4. To every presheaf we can associate an associated sheaf satisfying a universal property. This process is calledsheafification. Unlike for topological spaces, we cannot define the sheafification by considering functions on the disjoint union`
x∈XFxof stalks, since there might be sheaves w.r.t the Grothendieck topology that are nonzero yet have zero stalks at all points ofX. Instead, the construction relies on iterating the construction of a sheaf version of a Čech complex.
IfXis a set andTa Grothendieck topology on it, then we callXaG-topological space.
The case that is of interest to us is of course whenXis an affinoidK-space andTis some Grothendieck topology such that the open covers are defined using affinoid subdomains andOXis a sheaf with respect toT. There are two main variants of this:
Definition 4.5(Weak Grothendieck topology). LetXbe an affinoidK-space. Let CatT be the category of affinoid subdomains ofXwith the inclusions as morphisms. The set CovTconsists of all finite families(Ui→U)i∈Iof inclusions of affinoid subdomains of Xsuch thatU=S
i∈IUi. We callTtheweak Grothendieck topology onX.
Tate’s Acyclicity Theorem 3.2 states thatOX is a sheaf for the weak Grothendieck topology. To capture even more structure ofX, we can admit some more open sets and coverings:
Definition 4.6(Strong Grothendieck topology). LetXbe an affinoidK-space. Thestrong Grothendieck topology onXis given by the following data.
1. A subsetU⊆Xis called admissible open if there is a coveringU=S
i∈IUiofU by affinoid subdomainsUi⊆Xsuch that for all morphisms of affinoidK-spaces ϕ:Z→Xsatisfyingϕ(Z)⊆Uthe covering(ϕ−1(Ui))i∈IofZadmits a refinement that is a finite covering ofZby affinoid subdomains.
2. A covering V = S
j∈JVj of some admissible open subsetV ⊆ X by means of admissible open setsVj is called admissible if for each morphism of affinoidK- spacesϕ: Z → Xsatisfying ϕ(Z) ⊆ V the covering(ϕ−1(Vj))j∈J ofZadmits a refinement that is a finite covering ofZby affinoid subdomains.
It is necessary to show that the strong Grothendieck topology deserves its name.
Fortunately, this is not so hard.
Proposition 4.7. LetXbe an affinoidK-space. The strong Grothendieck topology is indeed a Grothendieck topology onX, satisfying the following conditions:
(G0) ∅andXare admissible open.
(G1) Let(Ui)i∈Ibe an admissible covering of an admissible open subsetU⊆X. Furthermore, letV ⊆Ube a subset such thatV∩Uiis admissible open for alli∈I. ThenVis admissible open inX.
(G2) Let(Ui)i∈Ibe a covering of an admissible open setU ⊆ Xby admissible open subsets Ui⊆Xsuch that(Ui)i∈Iadmits an admissible covering ofUas refinement. Then(Ui)i∈I itself is admissible.
We say that a morphismϕ:Z→Xof affinoidK-spaces equipped with Grothendieck topologies iscontinuousif the inverse imageϕ−1(U) of any admissible open subset U⊆Xis admissible open inZ, and the inverse image of any admissible covering inXis an admissible covering inZ. Now Proposition 2.11 shows that all morphisms between affinoidK-spaces are continuous with respect to the weak Grothendieck topologies. As one would hope, this extends to the strong topologies as well.
Proposition 4.8. LetY →Xbe a morphism of affinoidK-spaces. Thenϕis continuous with respect to the strong Grothendieck topologies onXandY.
In order for our definitions to be reasonable, the strong Grothendieck topology should also at least improve on the Zariski topology. This is indeed the case:
Lemma 4.9. LetXbe an affinoidK-space. Then the strong Grothendieck topology on Xis finer than the Zariski topology, i.e. every Zariski open subsetU⊆Xis admissible open and every Zariski covering is admissible.
LetTandT0be two Grothendieck topologies onXsuch that
• T0is a refinement ofT,
• eachT0-openU⊆Xadmits aT0-covering(Ui)i∈Iwhere allUiareT-open inX,
• eachT0-covering of aT-open subsetU⊆Xadmits aT-covering as a refinement Then we may uniquely (up to isomorphism) extend aT-sheafFonXto aT0-sheafF0. In factF0is given by
U7→lim−→
U
H0(U,F)
where the colimit is taken over allT0-coverings ofUconsisting ofT-open sets. Then one may check thatF0extendsFand is a sheaf as well.
Proposition 4.10. LetXbe an affinoidK-space. Then any sheafFonXwith respect to the weak Grothendieck topology admits a unique extension with respect to the strong Grothendieck topology.
In particular this is true for the presheafOXof affinoid functions.
This shows that there is a unique way to extend the sheafOXin the weak Grothendieck topology to the strong Grothendieck topology. The resulting sheaf is calledthe sheaf of rigid analytic functions onXand denotedOXas well. From now on, given an affinoid K-spaceX, we will use the strong Grothendieck topology and the sheaf of rigid analytic functions onX.
5 Rigid analytic spaces
Recall that aringedK-spaceis a pair(X,OX)whereXis a topological space andOXis a sheaf ofK-algebras onX.
Definition 5.1. AG-ringedK-spaceis a pair(X,OX)whereXis a G-topological space andOX is a sheaf ofK-algebras onX. If in addition all stalksOX,xforx∈ Xare local rings then(X,OX)is called alocally G-ringedK-space.
Morphisms of G-ringedK-spaces are defined as one would expect. One then constructs a functor from the category of affinoidK-spaces into locally G-ringedK-spaces and shows that it is fully faithful.
Definition 5.2. Arigid analyticK-spaceis a locally G-ringedK-space(X,OX)such that 1. the G-topology ofXsatisfies conditions(G0),(G1)and(G2)of Proposition 4.7, 2. Xadmits an admissible covering(Xi)i∈Iwhere(Xi,OX|Xi)is an affinoidK-space
for alli∈I.
Amorphism of rigid analyticK-spaces(X,OX)→(Y,OY)is a morphism in the sense of locally G-ringedK-spaces.
As done for schemes, one can show that rigid analyticK-spaces can be glued together from local pieces and transition morphisms. The same is true for morphisms between rigid analyticK-spaces. We also have:
Lemma 5.3. Fibered productsX×ZYexist in the category of rigid analyticK-spaces.
Since the usual definition of connectedness does not make sense for rigid analytic K-spaces, one supplants it by the following.
Definition 5.4. A rigid analytic K-spaceX is calledconnectedif there does not exist an admissible covering(X1,X2)ofXwhereX1,X2 ⊆Xare non-empty admissible open subspaces withX1∩X2=∅.
From Tate’s Acyclicity Theorem 3.2 it follows that an affinoidK-space Sp(A)is con- nected if and only if A cannot be written as a non-trivial cartesian product of two K-algebras.
6 GAGA
Similarly to the analytification functor overCthere is a “rigid analytification” functor over complete non-Archimedean fields.
Definition 6.1. Let(Z,OZ)be aK-scheme of locally finite type. Arigid analytification of(Z,OZ)is a rigidK-space(Zrig,OZrig)together with a morphism of locally G-ringed K-spacesι: (Zrig,OZrig)→(Z,OZ)satisfying the following universal property: Given a rigidK-space(Y,OY)and a morphism of locally G-ringedK-spaces(Y,OY)→(Z,OZ)the latter factors throughιvia a unique morphism of rigidK-spaces(Y,OY)→(Zrig,OZrig). Proposition 6.2. EveryK-schemeZlocally of finite type admits an analytificationZrig →Z.
Furthermore, the underlying map of sets identifies the points ofZrigwith the closed points ofZ. The universal property of rigid analytifications implies that we can also analytify morphisms betweenK-schemes.
Corollary 6.3. Rigid analytification defines a functor from the category ofK-schemes of locally finite type to the category of rigidK-spaces. It is called theGAGA functor.
It is worth noting that, just like in the complex case, even more is true. IfXis a proper K-scheme then the GAGA functor induces an equivalence between coherentOX-modules and coherentOXrig-modules (which we haven’t defined) and the cohomology groups agree. Furthermore, closed rigid subvarieties ofXrig arise as rigid analytifications of subvarieties ofXand all morphisms between rigidly analytifiedK-schemes are algebraic.
As an example, we will construct rigid affinen-spaceAn,rigK . The idea is to take the infinite union of balls of increasing radius and glue them together by the canonical maps.
Letr >0 andTn(r)be theK-algebra of all power seriesP
νaνζνwhereζ= (ζ1, . . . ,ζn) and limνaνr|ν|=0. This is the ring of all power series converging on the ball of radius r. Choose anyc∈Kwith|c|>1 and setTn(i)=Tn(|c|i). This algebra may be identified with the Tate algebraKhc−iζi. We have inclusions
K[ζ],→ · · ·,→Tn(2),→Tn(1),→Tn(0)=Tn (6.1) and this corresponds to inclusions of affinoid subdomains
Bn=SpTn(0),→SpTn(1),→SpTn(2),→ · · ·
We can interpret SpTn(i)as then-dimensional ball of radius|ci|. Taking the union of all SpTn(i)and gluing them along the inclusions yields the rigidK-spaceAn,rigK . It comes equipped with an admissible coveringAn,rigK =S∞
i=0SpTn(i).
Since intuitively the power series with limνaν∞ν= 0 are just the polynomials (i.e.
aν=0 for|ν|0) the following result is not surprising:
Lemma 6.4. The inclusions in (6.1) induce inclusions of maximal spectra MaxTn(0)⊆MaxTn(1)⊆ · · · ⊆MaxK[ζ]
and MaxK[ζ] =S∞
i=0MaxTn(i). This means that on the level of sets, the points ofAn,rigK correspond to the closed points ofAnK.
Similarly, we can construct the rigid analytification of any affineK-scheme of finite type. LetX=SpecK[ζ]/afor an ideala⊆K[ζ]. Then we consider the maps
K[ζ]/a→ · · · →Tn(2)/(a)→Tn(1)/(a)→Tn(0)/(a)
and this corresponds to inclusions of affinoid subdomains
SpTn(0)/(a),→SpTn(1)/(a),→SpTn(2)/(a),→ · · ·
and again we construct the rigid analytification ofXby gluing together all these pieces.
References
[Bos14] S. Bosch,Lectures on formal and rigid geometry. Springer, 2014 (cit. on p. 1).
