**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
to**admissible open subsets**and their**admissible open coverings. Sheaves on**Xwill 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 of**Berkovich spaces**is 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 =sup_{x∈Max}_{A}|f(x)|where MaxAis the set
of maximal ideals ofA.

**Definition 2.1.** LetAbe an affinoidK-algebra. The**affinoid**K-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.** A**morphism**σ: Sp(A) → Sp(B)**of affinoid**K**-spaces**is 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 the**Zariski topology**on 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 basisD_{f} 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.** The**canonical topology**ofXis the one generated by allX(f,ε).

We writeX(f) =X(f, 1)and

X(f_{1}, . . . ,f_{r}) =X(f_{1})∩ · · ·X(f_{r}) ={x∈X:|f_{i}(x)|61 for alli=1, . . . ,r}
forf_{1}, . . . ,f_{r}∈A. These sets are called**Weierstraß domains**inX.

**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) =Q_{n}

i=1(ζ−α_{i})inKand choose an embedding
A/x,→K. Then we have

ε=|f(x)| =|α_{i}|

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) −α_{i}|=max(|f(y)|,|α_{i}|)>|α_{i}|=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) −α_{i}|>ε^{n}

Hence if|h(y)| < ε^{n}then|f(y)| = ε. Taking anyc∈ K^{∗}with|c| < ε^{n}we setg = c^{−1}h

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.** *Let*ϕ: Sp(B)→Sp(A)*be a morphism of affinoid*K-spaces and letf_{1}, . . . ,f_{r}∈
A*. Then we have*

ϕ^{−1}(Sp(A)(f_{1}, . . . ,f_{r})) =Sp(B)(ϕ^{∗}(f_{1}), . . . ,ϕ^{∗}(f_{r}))
*In particular,*ϕ*is 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(f_{1}, . . . ,f_{r},g^{−1}_{1} , . . . ,g^{−1}_{s} ) =

x∈X:|f_{i}(x)|61,|g_{j}(x)|>1

**Laurent domains. Subsets of type**
X

f_{1}

f_{0}, . . . ,f_{r}
f_{0}

=

x∈X:|f_{i}(x)|6|f_{0}(x)|

forf_{0}, . . . ,f_{r}∈Awithout common zeros are called**rational 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 an**affinoid subdomain**ofX if there exists
a morphism of affinoidK-spaces ι:X^{0} → X such that ι(X^{0}) ⊆ Uand the following
universal property holds: Any morphism of affinoid K-spaces ϕ: Y → Xsatisfying
ϕ(Y)⊆Uadmits a unique factorization throughι:X^{0}→Xvia a morphism of affinoid
K-spacesϕ^{0}:Y→X^{0}.

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 idealm_{x} ⊆Awith residue fieldKand{x}as an affinoidK-space comes from the
algebraA/m_{x}. Then the inclusion{x},→Xcorresponds to the morphismA→A/m_{x}=K.

But we can also construct the mapϕ:A→A/m^{2}_{x}. 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 localizationA_{m}_{x}, 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(X^{0}) =U. Hence we can identify the setUwith the
set ofX^{0}and 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 affinoid*K*-space*X*are open*
*affinoid subdomains.*

*Proof.* We only prove this for a Weierstraß domainX(f)⊆X, wherefstands for a tuple
f_{1}, . . . ,f_{r} ∈A. Consider the affinoidK-algebraAhζ_{1}, . . . ,ζ_{r}iof restricted power series
and consider

Ahfi=Ahf_{1}, . . . ,f_{r}i=Ahζ_{1}, . . . ,ζ_{r}i/(ζ_{i}−f_{i})

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^{∗}(f_{i})(y)|=|f_{i}(ϕ(y))|

since we have an inclusionA/ϕ(y),→B/yof finite extensions ofK, induced byϕ^{∗}. Then
ϕ(Y)⊆X(f)if|ϕ^{∗}(f_{i})|sup 61 for alli=1, . . . ,r. Now considerϕ=ι. Hereι^{∗}(f_{i})is the
residue class ofζ_{i}inAhfi, hence its Gauß norm is 1. Since the supremum norm ofι^{∗}(f_{i})
is bounded by the residue norm, which in turn is bounded by the Gauß norm ofι^{∗}(f_{i}),
we have shownι(X^{0})⊆X(f).

We still have to show the universal property forι^{∗}, i.e. each morphismϕ^{∗}:A→ B
of affinoidK-algebras satisfying |ϕ^{∗}(f_{i})|sup 6 1 for all i = 1, . . . ,r admits a unique
factorization throughι^{∗}. We can extendϕ^{∗} to a morphism Ahζi → Bby mapping
ζ_{i} 7→ ϕ^{∗}(f_{i}). Then allζ_{i}−f_{i}belong 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 subdomains**special. Here are some results about properties**
of (special) affinoid subdomains that one would expect or hope to have.

**Proposition 2.11.** *Let*ϕ:Y → X*be a morphism of affinoid*K-spaces and letX^{0} ,→ X*be an*
*affinoid subdomain. Then*Y^{0}=ϕ^{−1}(X^{0})*is an affinoid subdomain of*Y*and we have*Y^{0}=Y×_{X}X^{0}*.*
*If*X^{0}*is a Weierstraß, Laurent or rational subdomain of*X*then the corresponding statement is*
*true for*Y^{0}*as an affinoid subdomain of*Y*. The defining functions of*X^{0}*pull back to the defining*
*functions of*Y^{0}*.*

*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 ⊆ X*be affinoid subdomains of* X*. Then* U∩V *is an affinoid*
*subdomain of*X*as well. Furthermore, if*U*and*V*are Weierstraß (resp. Laurent, resp. rational)*
*subdomains then the same is true for*U∩V.

**Proposition 2.13.** *Let*U → X*be a morphism of affinoid*K*-spaces defining*U*as an affinoid*
*subdomain of*X. ThenU*is open in*X*and the canonical topology of*X*restricts to the one of*U.

**Theorem 2.14** (Gerritzen–Grauert). *Let* X *be an affinoid*K-space andU ⊆ X*an affinoid*
*subdomain. Then*U*is a finite union of rational subdomains of*X.

**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 for*finite*coverings.

LetFbe a presheaf onXandU = (U_{i})_{i∈I}be a covering ofXby affinoid subdomains
U_{i}. We say thatFis aU-sheafif for all affinoid subdomainsU⊆Xthe sequence

OX(U)→Y

i∈I

OX(U_{i}∩U)⇒ Y

i,j∈I

OX(U_{i}∩U_{j}∩U)

is exact.

**Theorem 3.1**(Tate). *Let*X*be an affinoid*K*-space. The presheaf*OX*of affinoid functions is a*
U*-sheaf on*X*for all finite coverings*U = (U_{i})_{i∈I}*of*X*by affinoid subdomains*U_{i}⊆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 groupsH^{q}(U,F)vanish forq >0.

**Theorem 3.2**(Tate’s Acyclity Theorem). *Let*X*be an affinoid*K*-space and*U*be a finite covering*
*of*X*by affinoid subdomains. Then*U*is acyclic with respect to the presheaf*OX*.*

**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). A**Grothendieck topology**Tis a category CatT
together with a set CovTof families(U_{i}→U)_{i∈I}of morphisms in CatT, which we call
the**coverings, such that:**

1. Ifφ:U→Vis an isomorphism in CatTthen(φ)∈CovT.

2. If(U_{i}→U)_{i∈I}and(V_{ij}→U_{i})_{j∈J}_{i} are coverings, then(V_{ij}→U_{i}→U)_{i∈I,j∈J}_{i}is a
covering.

3. If (U_{i} → U)_{i∈I} is a covering and V → Ua morphism in CatTthen the fiber
productsU_{i}×_{U}Vexist in CatTand(U_{i}×_{U}V→V)_{i∈I}is 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 (U_{i})_{i∈I} of open subsets of U is an open cover ofU if the map
S

i∈IU_{i}→Uinduced by the inclusions is surjective.

Observe that forU_{1},U_{2}⊆Uopen sets we haveU_{1}×_{U}U_{2} =U_{1}∩U_{2}. 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(U_{i})_{i∈I}a
covering ofU, the family(U_{i}∩V)_{i∈I}is 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.

A**presheaf**onTwith values inCis a contravariant functorF: CatT→C. We say thatF
is a**sheaf**if

F(U)→Y

i∈I

F(U_{i})⇒ Y

i,j∈I

F(U_{i}×_{U}U_{j})

is exact for all coverings(U_{i}→U)_{i∈I}in CovT.

**Remark 4.4.** To every presheaf we can associate an associated sheaf satisfying a universal
property. This process is called**sheafification. 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 callXa**G-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(U_{i}→U)_{i∈I}of inclusions of affinoid subdomains of
Xsuch thatU=S

i∈IU_{i}. We callTthe**weak Grothendieck topology on**X.

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. The**strong**
**Grothendieck topology on**Xis given by the following data.

1. A subsetU⊆Xis called admissible open if there is a coveringU=S

i∈IU_{i}ofU
by affinoid subdomainsU_{i}⊆Xsuch that for all morphisms of affinoidK-spaces
ϕ:Z→Xsatisfyingϕ(Z)⊆Uthe covering(ϕ^{−1}(U_{i}))_{i∈I}ofZadmits a refinement
that is a finite covering ofZby affinoid subdomains.

2. A covering V = S

j∈JV_{j} of some admissible open subsetV ⊆ X by means of
admissible open setsV_{j} is called admissible if for each morphism of affinoidK-
spacesϕ: Z → Xsatisfying ϕ(Z) ⊆ V the covering(ϕ^{−1}(V_{j}))_{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.** *Let*X*be an affinoid*K-space. The strong Grothendieck topology is indeed a
*Grothendieck topology on*X*, satisfying the following conditions:*

(G_{0}) ∅*and*X*are admissible open.*

(G_{1}) *Let*(U_{i})_{i∈I}*be an admissible covering of an admissible open subset*U⊆X. Furthermore,
*let*V ⊆U*be a subset such that*V∩U_{i}*is admissible open for all*i∈I*. Then*V*is admissible*
*open in*X.

(G_{2}) *Let*(U_{i})_{i∈I}*be a covering of an admissible open set*U ⊆ X*by admissible open subsets*
U_{i}⊆X*such that*(U_{i})_{i∈I}*admits an admissible covering of*U*as refinement. Then*(U_{i})_{i∈I}
*itself is admissible.*

We say that a morphismϕ:Z→Xof affinoidK-spaces equipped with Grothendieck
topologies is**continuous**if 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.** *Let*Y →X*be a morphism of affinoid*K-spaces. Thenϕ*is continuous with*
*respect to the strong Grothendieck topologies on*X*and*Y.

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.

LetTandT^{0}be two Grothendieck topologies onXsuch that

• T^{0}is a refinement ofT,

• eachT^{0}-openU⊆Xadmits aT^{0}-covering(U_{i})_{i∈I}where allU_{i}areT-open inX,

• eachT^{0}-covering of aT-open subsetU⊆Xadmits aT-covering as a refinement
Then we may uniquely (up to isomorphism) extend aT-sheafFonXto aT^{0}-sheafF^{0}. In
factF^{0}is given by

U7→lim−→

U

H^{0}(U,F)

where the colimit is taken over allT^{0}-coverings ofUconsisting ofT-open sets. Then one
may check thatF^{0}extendsFand is a sheaf as well.

**Proposition 4.10.** *Let*X*be an affinoid*K*-space. Then any sheaf*F*on*X*with respect to the weak*
*Grothendieck topology admits a unique extension with respect to the strong Grothendieck topology.*

*In particular this is true for the presheaf*OX*of 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 called**the sheaf of**
**rigid analytic functions on**Xand 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 a**ringed**K**-space**is a pair(X,OX)whereXis a topological space andOXis a
sheaf ofK-algebras onX.

**Definition 5.1.** A**G-ringed**K-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 a**locally G-ringed**K**-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.** A**rigid analytic**K-spaceis a locally G-ringedK-space(X,OX)such that
1. the G-topology ofXsatisfies conditions(G_{0}),(G_{1})and(G_{2})of Proposition 4.7,
2. Xadmits an admissible covering(X_{i})_{i∈I}where(X_{i},OX|Xi)is an affinoidK-space

for alli∈I.

A**morphism of rigid analytic**K-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×_{Z}Yexist 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 called**connected**if there does not exist
an admissible covering(X_{1},X_{2})ofXwhereX_{1},X_{2} ⊆Xare non-empty admissible open
subspaces withX_{1}∩X_{2}=∅.

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. A**rigid analytification**
of(Z,OZ)is a rigidK-space(Z^{rig},OZ^{rig})together with a morphism of locally G-ringed
K-spacesι: (Z^{rig},O_{Z}^{rig})→(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)→(Z^{rig},OZ^{rig}).
**Proposition 6.2.** *Every*K-schemeZ*locally of finite type admits an analytification*Z^{rig} →Z.

*Furthermore, the underlying map of sets identifies the points of*Z^{rig}*with the closed points of*Z*.*
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 the**GAGA 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 coherentO_{X}^{rig}-modules (which we haven’t defined) and the cohomology groups
agree. Furthermore, closed rigid subvarieties ofX^{rig} arise as rigid analytifications of
subvarieties ofXand all morphisms between rigidly analytifiedK-schemes are algebraic.

As an example, we will construct rigid affinen-spaceA^{n,rig}_{K} . The idea is to take the
infinite union of balls of increasing radius and glue them together by the canonical maps.

Letr >0 andT_{n}(r)be theK-algebra of all power seriesP

νa_{ν}ζ^{ν}whereζ= (ζ_{1}, . . . ,ζ_{n})
and limνa_{ν}r^{|}^{ν}^{|}=0. This is the ring of all power series converging on the ball of radius
r. Choose anyc∈Kwith|c|>1 and setT_{n}^{(i)}=T_{n}(|c|^{i}). This algebra may be identified
with the Tate algebraKhc^{−i}ζi. We have inclusions

K[ζ],→ · · ·,→T_{n}^{(2)},→T_{n}^{(1)},→T_{n}^{(0)}=T_{n} (6.1)
and this corresponds to inclusions of affinoid subdomains

B^{n}=SpT_{n}^{(0)},→SpT_{n}^{(1)},→SpT_{n}^{(2)},→ · · ·

We can interpret SpT_{n}^{(i)}as then-dimensional ball of radius|c^{i}|. Taking the union of all
SpT_{n}^{(i)}and gluing them along the inclusions yields the rigidK-spaceA^{n,rig}_{K} . It comes
equipped with an admissible coveringA^{n,rig}_{K} =S_{∞}

i=0SpT_{n}^{(i)}.

Since intuitively the power series with limνa_{ν}∞^{ν}= 0 are just the polynomials (i.e.

a_{ν}=0 for|ν|0) the following result is not surprising:

**Lemma 6.4.** The inclusions in (6.1) induce inclusions of maximal spectra
MaxT_{n}^{(0)}⊆MaxT_{n}^{(1)}⊆ · · · ⊆MaxK[ζ]

and MaxK[ζ] =S_{∞}

i=0MaxT_{n}^{(i)}. This means that on the level of sets, the points ofA^{n,rig}_{K}
correspond to the closed points ofA^{n}_{K}.

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→ · · · →T_{n}^{(2)}/(a)→T_{n}^{(1)}/(a)→T_{n}^{(0)}/(a)

and this corresponds to inclusions of affinoid subdomains

SpT_{n}^{(0)}/(a),→SpT_{n}^{(1)}/(a),→SpT_{n}^{(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).*