Quotients Of Admissible Formal Schemes and Adic Spaces by Finite Groups
aa r X i v : . [ m a t h . AG ] F e b QUOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BYFINITE GROUPS
BOGDAN ZAVYALOV
Abstract.
In this paper we give a self-contained treatment of finite group quotients of admissible(formal) schemes and adic spaces that are locally topologically finite type over a locally stronglynoetherian adic space. Introduction
Overview.
This paper studies “geometric quotients” in different geometric setups. Namely,we work in three different situations: flat and locally finite type schemes over a typically non-noetherian valuation ring, admissible formal schemes over a complete microbial valuation ring (seeDefinition 3.1.1), and locally topologically finite type adic spaces over a locally strongly noetheriananalytic adic space. These 3 different contexts occupy Chapters 2, 3, and 4 respectively.The motivation to study these quotients comes from our paper [Zav20], where we (roughly) showthat any smooth rigid-space X over an algebraically closed non-archimedean field C locally has afinite ´etale covering f ′ : X ′ → X such that f ′ is a torsor for some finite group G , and X ′ admits a“polystable” admissible formal O C -model. We refer to [Zav20, Theorem 1.4] for a precise result. Inorder to formulate and prove this theorem, we had to make sure that a quotient of an admissibleformal O C -scheme by an O C -action of a finite group exists as an admissible formal O C -scheme.This result seems to be missing in the literature, the main difficulty being that the ring O C is nevernoetherian.1.1.1. Scheme Case.
Even though we are mostly interested in formal schemes and adic spaces, wenote that the question if
X/G is of finite type is already non-trivial for a flat, finite type (affine) O C -scheme X . To explain the main issue, we briefly recall what happens in the classical situationof a finite type R -scheme X with an R -action of a finite group G for some noetherian ring R . Undersome mild assumptions on X , one can rather easily reduce to the affine situation X = Spec A ,where the main work is to show that A G is of finite type over R . This is done in two steps: onefirstly checks that A is a finite A G -module, and then one uses the Artin-Tate Lemma: Lemma 1.1.1. [AM69, Proposition 7.8] Let R be a noetherian ring, and B ⊂ C an inclusion of R -algebras. Suppose that C is a finite type R -algebra, and C is a finite B -module. Then B isfinitely generated over R .One may think that probably the Artin-Tate lemma can hold, more generally, over a non-noetherian base R if C is finitely presented over R . However, this is not the case and the Artin-TateLemma fails over any non-noetherian base: This automatically implies finite presentation by Lemma 2.2.2. In particular, if X is quasi-projective over R . Example 1.1.2.
Let R be a non-noetherian ring with an ideal I that is not finitely generated.Consider the R -algebra C := R [ ε ] / ( ε ), and the R -subalgebra B = R ⊕ Iε . So C is a finitelypresented R -algebra, and C is finite as an B -module since it is already finite over R . However, B is not finitely generated R -algebra as that would imply that I is a finitely generated ideal.Example 1.1.2 shows that the strategy should be appropriately modified in the non-noetheriansituations like schemes over O C . We deal with this issue by proving a weaker version of the Artin-Tate Lemma over any valuation ring k + (see Lemma 2.2.3). That proof crucially exploits featuresof finitely generated algebras over a valuation ring. We emphasize that our argument does use the k + -flatness assumption in a serious way; we do not know if the quotient of a finitely presented affine k + -scheme by a finite group action is finitely presented (or finitely generated) over k + .1.1.2. Formal Schemes and Adic Spaces.
The strategy above can be appropriately modified to workin the world of admissible formal schemes and strongly noetherian adic spaces. In both situations,the main new input is a corresponding version of the Artin-Tate Lemma (see Lemma 3.2.3 andLemma 4.2.4). However, there are issues that are not seen in the scheme case. We explain a few ofthe main new technical difficulties that arise while proving the result in the world of adic spaces.The underlying topological space of an affinoid space Spa(
A, A + ) is harder to express in termsof the pair ( A, A + ). It is a set of all valuations on A with corresponding continuity and integralityconditions. In particular, even if one works with rigid spaces over a non-archimedean field K ,one has to take into account points of higher rank that do not have any immediate geometricmeaning. Hence, it takes extra care to identify Spa( A G , A + ,G ) with Spa( A, A + ) /G even on thelevel of underlying topological spaces .Furthermore, the notion of a topologically finite type (resp. finite) morphism of Tate-Huber pairsis more subtle than its counterpart in the algebraic setup for two different reasons. Firstly, it hasa topological aspect that takes some care to work with. Secondly, the notion involves conditionson both A and A + (see Definition B.2.1 and Definition B.2.6). Usually, A + is non-noetherian, so itrequires some extra work to check the relevant condition on it.1.1.3. Generality.
In the case of adic spaces, we consider spaces that are locally topologically finitetype over a strongly noetherian analytic adic space in Section 4. One reason for this level ofgenerality is to include adic spaces that are topologically finite type over Spa( k, k + ) for a microbialvaluation ring k + (see Definition 3.1.1). These spaces naturally arise while studying fibers ofmorphisms of rigid spaces X → Y over points of Y of higher rank . We think that the categoryof strongly noetherian analytic adic space is the natural one to consider. One of its advantagesis that it contains both topologically finite type morphisms and morphisms coming from the (notnecessary finite) base field extension in rigid geometry.In the case of formal schemes, the results of Section 3 are written in the generality of admissibleformal schemes over a complete, microbial valuation ring k + (see Definition 3.1.1). We want topoint out that Appendix A contains versions of the main results of Section 3 for a topologicallyuniversally adhesive base (see Definition A.3.12). These results are more general and include both the cases of formal schemes topologically finite type over some k + and noetherian formal schemes.However, we prefer to formulate and prove the results in the main body of the paper for admissibleformal schemes over k + since it simplifies the exposition a lot. We only refer to Appendix A for thenecessary changes that have to be made to make the arguments work in the more general adhesivesituation. Considered as adic spaces
UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 3
Likewise, Appendix A has versions of the results of Section 2 over a universally adhesive base(see Definition A.2.1). But we want to point out that a valuation ring k + is universally adhesiveonly if it is microbial (see Lemma A.2.3), so the results of Appendix A do not fully subsume theresults of Section 2.2.1.2. Comparison with [Han].
While writing this paper, we found that similar results for adicspaces were already obtained in [Han]. We briefly discuss the main similarities and differences inour approaches.David Hansen separately discusses two different situations: rigid spaces over a non-archimedeanfield K , and general analytic adic spaces. In the former case, he shows that (under some assumptionson X ) X/G exists as a rigid space over K for any finite group G . He crucially uses [BGR84,Proposition 6.3.3/3] that states that for a K -affinoid affinoid A with a K -action of a finite group G , the ring of invariants A G is a K -affinoid algebra. The proof of this result uses analytic input:the Weierstrass preparation theorem. In the latter case, he shows that the quotient of X exists asan analytic adic space if the order of G is invertible in O X ( X ). The argument there is based onan averaging trick, so it uses the invertibility assumption in order to be able to divide by G . Wenote that if X is a perfectoid space over a perfectoid field, he can drop this invertibility assumptionby some other argument. The whole point of the latter case is to be able to work with “big” adicspaces such as perfectoid spaces.In contrast with Hansen’s approach, our methods neither use any non-trivial input from non-archimedean analysis, nor the averaging trick. What we do is try to imitate the classical algebraicargument based on the Artin-Tate Lemma in the setup of strongly noetherian adic spaces. Moreprecisely, we show that if X is a locally topologically finite type adic space (with some otherconditions) over a locally strongly noetherian adic space S with an S -action of a finite group G then the quotient X/G exists as a locally topologically finite type adic S -space. Our result does notrecover Hansen’s result as we do not allow “big” adic spaces such as perfectoid spaces, but it provesa stronger statement in the case of adic spaces locally of finite type over a strongly noetherian Tateaffinoid as we do not have any assumptions on the order of G . Moreover, even in the case of rigidspaces, it gives a new proof of the existence of X/G as a rigid space that does not use much ofanalytic theory.1.3.
Our results.
We firstly study the case of a flat, locally finite type scheme X over a valuationring k + and a k + -action of a finite group G . We show that X/G exists as a flat, locally finite type k + -scheme under a mild assumption on X : Theorem 1.3.1. (Theorem 2.1.16 and Theorem 2.2.6) Let X be a flat, locally finite type k + -schemewith a k + -action of a finite group G . Suppose that each point x ∈ X admits affine neighborhood V x containing G.x . Then
X/G exists as a flat, locally finite type k + -scheme. Moreover, it satisfiesthe following properties:(1) π : X → X/G is universal in the category of G -invariant morphisms to locally ringed S -spaces.(2) π : X → X/G is a finite, finitely presented morphism (in particular, it is closed).(3) Fibers of π are exactly the G -orbits.(4) The formation of the quotient X/G commutes with flat base change (see Theorem 2.1.16(4)for the precise statement). Defined as locally topologically finite type adic spaces over Spa( K, O K ) BOGDAN ZAVYALOV
We then consider quotients of admissible formal schemes X over a complete microbial valuationring k + by a k + -action of a finite group G . Under similar conditions, we show that X /G exists asan admissible formal k + -scheme and satisfies the expected properties: Theorem 1.3.2. (Theorem 3.3.4) Let X be an admissible formal k + -scheme with a k + -action of afinite group G . Suppose that each point x ∈ X admits an affine neighborhood V x containing G.x .Then X /G exists as an admissible formal k + -scheme. Moreover, it satisfies the following properties:(1) π : X → X /G is universal in the category of G -invariant morphisms to topologically locallyringed spaces over S .(2) π : X → X /G is a surjective, finite, topologically finitely presented morphism (in particular,it is closed).(3) Fibers of π are exactly the G -orbits.(4) The formation of the geometric quotient commutes with flat base change (see Theorem 3.3.4(4)for the precise statement).Finally, we consider the case of locally topologically finite type adic spaces over a locally stronglynoetherian adic space. Theorem 1.3.3. (Theorem 4.3.4) Let S be a locally strongly noetherian analytic adic space (seeDefinition B.2.15), and X a locally topologically finite type adic S -space with an S -action of afinite group G . Suppose that each point x ∈ X admits affinoid open neighborhood V x containing G.x . Then
X/G exists as a locally topologically finite type adic S -space. Moreover, it satisfies thefollowing properties:(1) π : X → X/G is universal in the category of G -invariant morphisms to valuative topologi-cally locally ringed S -spaces.(2) π : X → X/G is a finite, surjective morphism (in particular, it is closed).(3) Fibers of π are exactly the G -orbits.(4) The formation of the geometric quotient commutes with flat base change (see Theorem 4.3.4(4)for the precise statement).The natural question is whether these quotients commute with certain functors like formal com-pletion, analytification, and adic generic fiber. We show that this is indeed the case, i.e. theformation of the geometric quotients commute with the mentioned above functors whenever theyare defined. We informally summarize the results below: Theorem 1.3.4. (Theorem 3.4.1, Theorem 4.4.1, and Theorem 4.5.3)(1) Let k + be a microbial valuation ring, and X a flat, locally finite type k + -scheme with a k + -action of a finite group G . Suppose X satisfies the assumption of Theorem 1.3.1. Thenatural morphism b X/G → [ X/G is an isomorphism.(2) Let k + be a complete, microbial valuation ring with fraction field k , and X an admissibleformal k + -scheme with a k + -action of a finite group G . Suppose X satisfies the assumptionof Theorem 1.3.2. The natural morphism X k /G → ( X /G ) k is an isomorphism.(3) Let K be a complete rank-1 valued field, and X a locally finite type K -scheme with a K -action of a finite group G . Suppose X satisfies the assumption of Theorem 2.1.16. Thenatural morphism X an /G → ( X/G ) an is an isomorphism. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 5
Acknowledgements.
We are grateful to B. Bhatt, B. Conrad and S. Petrov for fruitful conver-sations. We express additional gratitude to B. Conrad for reading the first draft of this paper andmaking lots of suggestions on how to improve the exposition of this paper.2.
Quotients of Schemes
Review of Classical Theory.
We review the classical theory of quotient of schemes by anaction of a finite group. This theory was developed in [SGA 1, Exp. V, § S . Definition 2.1.1.
Let G be a finite group, and X a locally ringed space over S with a right S -action of G . The geometric quotient X/G = ( | X/G | , O X/G , h ) consists of: • the topological space | X/G | := | X | /G with the quotient topology. We denote by π : | X | →| X/G | the natural projection, • the sheaf of rings O X/G := ( π ∗ O X ) G , • the morphism h : X/G → S defined by the pair ( h, h ), where h : | X | /G → S is the uniquemorphism induced by f : X → S and h is the natural morphism O S → h ∗ (cid:0) O X/G (cid:1) = h ∗ (cid:16) ( π ∗ O X ) G (cid:17) = ( h ∗ ( π ∗ O X )) G = ( f ∗ O X ) G that comes from G -invariance of f . Remark 2.1.2.
We note that
X/G is, a priori, only a ringed space. However, Lemma 2.1.3 showsthat
X/G is actually a locally ringed space and π : X → X/G is a morphism of locally ringedspaces.
Lemma 2.1.3.
Let X be a locally ringed space over S with a right S -action of a finite group G . Then X/G is a locally ringed space, and π : X → X/G is a map of locally ringed spaces (so
X/G → S is too). Remark 2.1.4.
This lemma must be well-known, but we do not know any particular reference.We decided to include the proof as it will be a convenient technical tool for us.Lemma 2.1.3 allows us to construct quotients entirely in the category of locally ringed spacesand not merely in the category of all ringed spaces. The main technical issue with the category ofringed spaces is that locally ringed spaces do not form a full subcategory of it.
Proof.
We firstly describe the stalk O X/G,x for a point x ∈ X/G with a lift x ∈ X . The constructionof X/G implies that O X/G,x ≃ colim { x ∈ U ⊂ X | g ( U )= U ∀ g ∈ G } O X ( U ) G . (1)Now we note that G -stable opens of x ∈ X are cofinal among all opens in X around x . Thus,we can write O X,x as the colimit over G -stable opens; i.e. O X,x ≃ colim { x ∈ U ⊂ X | g ( U )= U ∀ g ∈ G } O X ( U ) . (2)Therefore, the natural map O X/G,x → O X,x is an inclusion. We want to show that it is a local map of local rings. This is equivalent to say that m x ∩ O X/G,x is the unique maximal ideal in O X/G,x or, equivalently, that any f ∈ O X/G,x ∩ O × X,x lies in O × X/G,x . BOGDAN ZAVYALOV
We use (1) and (2) to find a G -stable open x ∈ U ⊂ X such that f comes from an element f U ∈ O X ( U ) G and such that a multiplicative inverse f − U ∈ O X ( U ) exists. The uniqueness ofmultiplicative inverses implies that f − U is G -invariant. This means that f ∈ O × X/G,x . (cid:3) Remark 2.1.5.
It is trivial to see that the pair (
X/G, π ) is a universal object in the category of G -invariant morphisms to locally ringed spaces over S . Remark 2.1.6.
We warn the reader that
X/G might not be a scheme even if S = Spec C and X is a smooth and proper, connected C -scheme with a C -action of G = Z / Z . Namely, Hironaka’sexample [Ols16, Example 5.3.2] is smooth and proper, connected 3-fold over C with a C -action of Z / Z such that there is an orbit G.x that is not contained in any open affine subscheme U ⊂ X .Lemma 2.1.8 below implies that X/G is not a scheme.
Lemma 2.1.7.
Let X be an S -scheme with an S -action of a finite group G . Suppose that eachpoint x ∈ X admits an open affine subscheme V x that contains the orbit G.x . Then the same holdswith X replaced by any G -stable open subscheme U ⊂ X . Proof.
Let x be a point in U , and V x an open affine in X that contains G.x . Consider W x := U ∩ V x that is an open (possibly non-affine) neighborhood of x ∈ U containing G.x . It suffices to show astronger claim that any finite set of points in W x is contained in an open affine. This follows from[EGA II, Corollaire 4.5.4] as W x is an open subscheme inside the affine scheme V x . (cid:3) Lemma 2.1.8.
Let R be a noetherian ring, and X a separated, finite type R -scheme with an R -action of a finite group G . Suppose that there is a point x ∈ X such that the orbit G.x is notcontained in any open affine subscheme U ⊂ X . Then X/G is a not a scheme.
Remark 2.1.9.
Lemma 2.1.8 must have been known to experts for a long time. However, we arenot aware of any reference for this fact. For example, [SGA 1, Exp. V, Proposition 1.8] shows thatit is impossible for
X/G to be a scheme and for π : X → X/G to be affine . We strengten the resultand show that X/G is not a scheme without the affineness requirement on π . Proof.
Suppose that
X/G is an R -scheme, and consider the image x := π ( x ) ∈ X/G . It admitsan affine neighborhood U ⊂ X/G ; this defines an open G -stable subscheme U := π − ( U ) ⊂ X containing the orbit G.x .Now we note that the morphism π | U : U → U is quasi-finite and separated. Indeed, it is separatedof finite type since U is separated of finite type over R and U is separated; its fibers are finite bythe construction. Therefore, Zariski’s main theorem [EGA IV , Proposition 18.12.12] implies that π | U is quasi-affine, i.e. the natural morphism U → Spec O U ( U )is a quasi-compact open immersion. We note that Spec O U ( U ) naturally admits an action of thegroup G induced by the action of G on O U . Trivially, any point y ∈ Spec O U ( U ) admits an affineneighborhood containing G.y . Thus, Lemma 2.1.7 applied to Spec O U ( U ) and its open subscheme U implies that the same holds for U . As the result, the orbit G.x is contained in some open affinesubscheme of X . (cid:3) Definition 2.1.1 is useless unless we can verify that
X/G is a scheme if X is. The main goal ofthe rest of the section is to review when this is the case under some (mild) assumptions on X . To use this result, we recall that the structure sheaf O Y is ample on any affine scheme Y . Affineness of X → X/G is part of the definition an “admissible” action of G on X introduced in [SGA 1, Exp.V, Definition 1.7]. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 7
We start with the case of an affine scheme X = Spec A and an affine scheme S = Spec R . Thenthe natural candidate for the geometric quotient is Y = Spec A G . There is an evident G -invariant S -map p : X → Y that induces a commutative triangle XX/G Y. π pφ
We wish to show that φ is an isomorphism. Before doing this, we need to recall certain (well-known)properties of G -invariants. We include some proofs for the convenience of the reader. Lemma 2.1.10.
Let A be an R -algebra with an R -action of a finite group G . Then(1) the inclusion A G → A is integral. In particular, the morphism Spec A → Spec A G is closed.(2) Spec A → Spec A G is surjective, the fibers are exactly G -orbits.(3) If A is of finite type over R . Then A G → A is finite. Proof.
This is [SGA 1, Expose V, Proposition 1.1(i), (ii) and Corollaire 1.5]. We also point outthat the results follow from [AM69, Exercise 5.12, 5.13], and the observation that an integral, finitetype morphism is finite. (cid:3)
Remark 2.1.11.
We warn the reader that Lemma 2.1.10 does not imply that A G is of finite typeover R since we allow non-noetherian R (as needed later). Lemma 2.1.12.
Let R be a ring, and A an R -algebra with an action of a finite group G . Then theformation of invariants A G commutes with flat base change, i.e. for any flat R -algebra morphism A G → B the natural homomorphism B → ( B ⊗ A G A ) G is an isomorphism. Proof.
The proof is outlined just after [SGA 1, Exp. V, Proposition 1.9]. (cid:3)
Proposition 2.1.13.
Let X = Spec A be an affine R -scheme with an R -action of a finite group G .Then the natural map φ : X/G → Y = Spec A G is an R -isomorphism of locally ringed spaces. Inparticular, X/G is an R -scheme. Proof.
This is shown in [SGA 1, Exp. V, Proposition 1.1]. We review this argument here as thistype of reasoning will be adapted to more sophisticated situtions later in the paper.
Step 1. φ is a homeomorphism : We note that Lemma 2.1.10 ensures that p : X → Spec A G is aclosed, surjective map with fibers being exactly G -orbits. Thus, π : X → X/G and p : X → Spec A G are both topological quotient morphisms with the same fibers (namely, G -orbits). So the inducedmap f is clearly a homeomorphism. Step 2. φ is an isomorphism of locally ringed spaces : We use Lemma 2.1.10 again to check thatthe morphism of sheaves φ : O Y → φ ∗ O X/G is an isomorphism. Using the base of basic affineopens in Y , it suffices to show that the map( A G ) f → ( A f ) G ≃ ( A ⊗ A G A Gf ) G is an isomorphism for any f ∈ A G . This follows from Lemma 2.1.12 as ( A G ) f is A G -flat. (cid:3) Now we want to discuss when
X/G exists as a scheme in the global set-up without a separatednessassumption. Roughly, we want to cover X by G -stable affines and then deduce the claim fromProposition 2.1.13. In order to do this, we need the following lemma: BOGDAN ZAVYALOV
Lemma 2.1.14.
Let X be an S -scheme with an S -action of a finite group G . Suppose that forany point x ∈ X there is an open affine subscheme V x ⊂ X that contains the orbit G.x . Then eachpoint x ∈ X has a G -stable open affine neighborhood U x ⊂ X . Proof.
The proof is outlined just after [SGA 1, Exp V, Proposition 1.8], we recall the key stepshere. Firstly, Lemma 2.1.7 ensures that one can reduce to the case of an affine base an S = Spec R .Then one shows the claim for a separated X , in which case U x := T g ∈ G g ( V x ) is affine and does thejob. In general, Lemma 2.1.7 guarantees that one can replace X with the separated open subscheme T g ∈ G g ( V x ) and reduce to the separated case. (cid:3) We recall one case when the condition of Lemma 2.1.14 is satisfied.
Proposition 2.1.15.
Let φ : X → S be a locally quasi-projective S -scheme with an S -action of afinite group G . Then every point x ∈ X admits an affine neighborhood containing the orbit G.x . Proof.
The statement is local on S , so we may and do assume that S = Spec R is affine and thereis a quasi-compact R -immersion X ⊂ P NR . Then it suffices to show a stronger claim that any finiteset of points is contained in an open affine. This is shown in [EGA II, Corollaire 4.5.4]. (cid:3) Now, we are ready to explain the main existence result [SGA 1, Exp V, Proposition 1.8]. Forlater needs, we give a slightly different proof.
Theorem 2.1.16.
Let X be an S -scheme with an S -action of a finite group G . Suppose that eachpoint x ∈ X admits affine neighborhood V x containing G.x . Then
X/G is an S -scheme. Moreover,it satisfies the following properties:(1) π : X → X/G is universal in the category of G -invariant morphisms to locally ringed S -spaces.(2) π : X → X/G is an integral, surjective morphism (in particular, it is closed). The morphism π is finite if X is locally of finite type over S .(3) Fibers of π are exactly the G -orbits.(4) The formation of the geometric quotient commutes with flat base change, i.e. for any flatmorphism Z → X/G , the geometric quotient ( X × X/G Z ) /G is a scheme, and the naturalmorphism ( X × X/G Z ) /G → Z is an isomorphism. Proof. Step 1.
X/G is an S -scheme : We note that the claim is local on S , so we can use Lemma 2.1.7to reduce to the case S is affine. Now Lemma 2.1.14 allows to cover X by G -stable open affinesubschemes U i . Then the construction of the geometric quotient implies that π ( U i ) ⊂ X/G is an open subset that is naturally isomorphic to U i /G , and π − ( U i /G ) coincides with U i . Thisimplies that it suffices to show that U i /G is a scheme. This was already shown in Proposition 2.1.13. Step 2. π : X → X/G is surjective, integral (resp. finite) and fibers are exactly the G -orbits :Similar to Step 1, we can assume that X and S are affine. Then apply Lemma 2.1.10. Step 3. π : X → X/G is universal and commutes with flat base change : The universality isessentially trivial (Remark 2.1.5). To show the latter claim, we can again assume that X = Spec A and S = Spec R are affine and it suffices to consider affine Z . Then the claim follows fromLemma 2.1.12 and the identification of X/G with Spec A G . (cid:3) I.e. there exists an open covering S = ∪ V j such that each φ − ( V j ) → V j factors through a quasi-compactimmersion φ − ( V j ) → P NV j for some N . UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 9
Schemes Over a Valuation Ring k + . The main drawback of Theorem 2.1.16 is that if R is not noetherian we do not know if X/G is finite type over S = Spec R when X is. This makesthis theorem difficult to apply in practice. The main work is to show that a ring of invariants A G is finite type over R is A is. If R is noetherian, this problem is resolved using the Artin-TateLemma 1.1.1. The main goal of this section is to generalize it to certain non-noetherian situations.For the rest of the section, we fix a valuation ring k + with fraction field k and maximal ideal m k . Definition 2.2.1.
Let N ⊂ M be an inclusion of k + -modules, we say that N is saturated in M ifthe quotient M/N is k + -torsion free. Lemma 2.2.2.
Let k + be a valuation ring, A a finite type k + -algebra, and M a finite A -module.Then(1) A k + -module N is flat over k + if and only if it is torsion free.(2) If M is k + -flat, it is a finitely presented A -module.(3) If A is k + -flat, it is a finitely presented k + -algebra.(4) Let N ⊂ M be a saturated A -submodule of M . Then N is a finite A -module. Proof.
By [Mat86, Theorem 7.7] a k + -module is flat if and only if I ⊗ k + N → N is injective for any finitely generated ideal I ⊂ k + . But such I is principal since k + is a valuation ring, so we are done.The second and third claims are proven in [Sta19, Tag 053E].Now we show the last claim. We consider the quotient module M/N . The saturatedness as-sumption says that it is k + -flat, and it is clearly finite as an A -module. Thus, (2) ensures that M/N is finitely presented over A . So N is a finite A -module as it is the kernel of a homomorphismfrom a finite module to a finitely presented one. (cid:3) Lemma 2.2.3 (Non-noetherian Artin-Tate) . Let A → B be a finite injective morphism of k + -algebras. Suppose that B is a finite type k + -algebra and A is a saturated k + -submodule of B (inthe sense of Definition 2.2.1). Then A is a k + -algebra of finite type. Proof.
By the assumption, B is of finite type over k + , so there is a finite set of elements x i ∈ B such that the k + -algebra homomorphism p : k + [ T , . . . , T n ] → B that sends T i to x i is surjective. Since B is a finite A -module, we can choose some A -modulegenerators y , . . . , y m ∈ B . The choice of x , . . . , x n and y , . . . , y m implies that there are some a i,j , a i,j,l ∈ A with the relations x i = X j a i,j y i y i y j = X l a i,j,l y l . Now consider the k + -subalgebra A ′ of A generated by all a i,j and a i,j,l . Clearly, A ′ is of finite typeover k + . Moreover, B is finite over A ′ as y , . . . , y m are A ′ -module generators of B .We use Lemma 2.2.2(4) over A ′ to ensure that A is finite over A ′ as it is a saturated A ′ -submoduleof the finite A ′ -module B . Therefore, A is of finite type over k + . (cid:3) Corollary 2.2.4.
Let A be a flat, finite type k + -algebra with a k + -action of a finite group G . The k + -flat A G is a finite type k + -algebra, and the natural morphism A G → A is finitely presented. Proof.
Lemma 2.1.10 gives that A is a finite A G -module, and by k + -flatness of A clearly A G is asaturated k + -submodule of A . Therefore, Lemma 2.2.3 implies that A G is flat and finite type over k + . Now Lemma 2.2.2 (2) ensures that A is a finitely presented A G -module as it is A G -finite and k + -flat. Thus, it is finitely presented as an A G -algebra by [EGA IV , Proposition 1.4.7]. (cid:3) Remark 2.2.5.
Lemma 2.2.3 and Corollary 2.2.4 have versions over a universally adhesive base(see Definition A.2.1). We refer to Lemma A.2.5 and Corollary A.2.6 for the precise results.
Theorem 2.2.6.
Let X be a flat, locally finite type k + -scheme with a k + -action of a finite group G . Suppose that each point x ∈ X admits an affine neighborhood V x containing G.x . Then thescheme
X/G as in Theorem 2.1.16 is flat and locally finite type over k + , and the integral surjection π : X → X/G is finite and finitely presented.
Proof of Theorem 2.2.6.
By construction,
X/G is clearly k + -flat. To show that X/G is locally offinite type and that π is finitely presented, we reduce to the affine case by passing to a G -stableaffine open covering of X (see Lemma 2.1.14). Now apply Corollary 2.2.4. (cid:3) Remark 2.2.7.
Theorem 2.2.6 also has a version for the base scheme S a universally I -adicallyadhesive for some quasi-coherent ideal of finite type I (see Definition A.2.7). We refer to Theo-rem A.2.9. 3. Quotients of Admissible Formal Schemes
We discuss the existence of quotient of some class of formal schemes by an action of a finitegroup G . The strategy to construct the quotient spaces is close to the one used in Section 2. Wefirstly construct the candidate space X /G that is, a priori, only a topologically locally ringed space.This construction clearly satisfies the universal property, but it is not clear (and generally false) if X /G is a formal scheme. We resolve this issue by firstly showing that it is a formal scheme if X isaffine. Then we argue by gluing to prove the claim for a larger class of formal schemes.There are two main complications compared to Section 2. The first one is that we cannot anymorefirstly show that X /G is a scheme by a very general argument and then study its properties underfurther assumptions, e.g. show that it is flat or (topologically) finite type over the base. The problemcan be seen even in the case of an affine formal scheme X = Spf A . The proof of Proposition 2.1.13crucially uses that the localization A Gf is A G -flat for any f ∈ A G . The analog in the world of formalschemes would be that the completed localization A G { f } = c A Gf is A G -flat. However, this requires somefiniteness assumption on A G in order to hold. Therefore, we need to verify algebraic properties of A G at the same as constructing the isomorphism Spf A/G ≃ Spf A G .The second, related problem is that one needs to be more careful with certain topological aspectsof the theory. For instance, the fiber product of affine formal schemes is given by the completed tensor product on the level of correspoding algebras. This is a more delicate functor as it is neitherleft nor right exact. So we pay extra attention to make sure that these complications do not causeany issues under suitable assumptions.3.1. The Setup and the Candidate Space X /G .Definition 3.1.1. A valuation ring k + is microbial if it has a finitely generated (hence principal)ideal of definition I , i.e. any neighborhood 0 ∈ U ⊂ k + open in the valuation topology contains I n for some n . UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 11
Definition 3.1.2.
An element ̟ ∈ k + is a pseudo-uniformizer if ( ̟ ) ⊂ k + is an ideal of definitionin k + . Example 3.1.3.
Any valuation ring k + of finite rank is microbial. This follows from the charac-terization of microbial valuation in [Sem15, Proposition 9.1.3(3)].More generally, a valuation ring k ( x ) + ⊂ k ( x ) associated with any point x ∈ X of an analytic adicspace (see Definition B.1.3) X is microbial. This can be seen from [Sem15, Proposition 9.1.3(2)].For the rest of the section, we fix a complete, microbial valuation ring k + with a pseudo-uniformizer ̟ . We denote by S the formal spectrum Spf k + .A formal k + -scheme will always mean an ̟ -adic formal k + -scheme. It is easy to see that thisnotion does not depend on a choice of an ideal of definition. Definition 3.1.4. A k + -algebra A is called admissible if A is k + -flat and topologically of finitetype (i.e. there is a surjection k + h t , . . . t d i → A ).A formal k + -scheme X is called admissible if it is k + -flat and locally topologically of finite type. Remark 3.1.5. (1) We note that there are many (non-equivalent) ways to define flatness informal geometry. They are all equivalent for a morphism f : X → Y of locally topologicallyfinite type formal k + -schemes.We prefer to use the following as the definition: f is flat if f f ( x ) : O Y ,f ( x ) → O X ,x is flatfor all x ∈ X (i.e. f is flat as a morphism of locally ringed spaces). We mention that in thecase f : Spf B → Spf A a morphism of affine, topologically finite type formal k + -schemes,this notion is equivalent to the flatness of A → B . This follows from [FK18, PropositionI.4.8.1 and Corollary I.4.8.2] and Remark A.3.3 (see [FK18, § I.2.1(a)] to relate adhesivenessto rigid-noetherianness).(2) Similarly, a morphism f : Spf B → Spf A of formal k + -schemes is topologically of finitetype if and only if A → B is topologically of finite type (see [FK18, Lemma I.1.7.3]).(3) In particular, if X = Spf A is an admissible formal k + -scheme, the k + -algebra A is admis-sible.We summarize the main properties of locally topologically finite type formal k + -schemes in thelemma below: Lemma 3.1.6.
Let k + be a complete, microbial valuation ring, A be a topologically finite type k + -algebra, and M a finite A -module. Then(1) M is ̟ -adically complete. In particular, A is ̟ -adically complete.(2) If A is k + -flat, it is topologically finitely presented.(3) If M is k + -flat, it is finitely presented over A .(4) Let N ⊂ M be a saturated (in the sense of Definition 2.2.1) A -submodule of M . Then N is a finite A -module.(5) Let N ⊂ M be an A -submodule of M . Then the ̟ -adic topology on M restricts to the ̟ -adic topology on N .(6) For any element f ∈ A , the completed localization A { f } = lim n A f /̟ n A f is A -flat. Proof.
The first claim is [Bos14, Proposition 7.3/8]. The second is [Bos14, Corollary 7.3/5]. Thethird in [Bos14, Theorem 7.3/4]. The fourth and fifth are covered by [Bos14, Lemma 7.3/7]. Andthe last follows from [Bos14, Proposition 7.3/11] and [Sta19, Tag 05GG]. (cid:3)
Definition 3.1.7.
Let G be a finite group, and X a locally topologically ringed space over S witha right S -action of G . The geometric quotient X /G = ( | X /G | , O X /G , h ) consists of: • the topological space | X /G | := | X | /G with the quotient topology. We denote by π : | X | →| X /G | the natural projection. • the sheaf of topological rings O X /G := ( π ∗ O X ) G with the subspace topology. • the morphism h : X /G → S defined by the pair ( h, h ), where h : | X | /G → S is the uniquemorphism induced by f : X → S and h is the natural morphism O S → h ∗ (cid:0) O X /G (cid:1) = h ∗ (cid:16) ( π ∗ O X ) G (cid:17) = ( h ∗ ( π ∗ O X )) G = ( f ∗ O X ) G that comes from G -invariance of f . Remark 3.1.8.
We note that Lemma 2.1.3 ensures that X /G is a topologically locally ringed S -space, and π : X → X /G is a morphism of topologically locally ringed S -paces (so X /G → S istoo). It is trivial to see that the pair ( X /G, π ) is a universal object in the category of G -invariantmorphisms to topologically locally ringed S -spaces.Our main goal is to show that under some mild assumptions, X /G is an admissible formal S -scheme when X is. We start with the case of affine formal schemes and then move to the generalcase.3.2. Affine Case.
We show that the quotient X /G of an admissible affine formal k + -scheme X =Spf A is canonically isomorphic to Spf A G that is, in turn, an admissible formal k + -scheme. Wepoint out that in constrast with the scheme case, we need firstly to establish that A G is an admissible k + -algebra, and only then we can show that X /G is isomorphic to Spf A G . Therefore, we start thesection with studying certain properties of the ring of invariants A G . Lemma 3.2.1.
Let (
R, I ) be a ring with a finitely generated ideal I , and A be an I -adicallycomplete R -algebra with an R -action of a finite group G . Then A G is complete in the I -adictopology. Proof.
Clearly A G is I -adically separated since A is, and it is complete for the subspace topologyfrom A . But all this means by design is that A G → lim n A G / ( I n A ∩ A G )is an isomorphism, and we need to justify that this implies that A G → lim n A G /I n A G is an isomorphism. For this purpose we will use that I is finitely generated.We already know that A G → lim n A G /I n A G is injective as A G is I -adically separated. So wenow show surjectivity. By [Sta19, Tag 090S] (which uses that I is finitely generated), it suffices tojustify surjectivity of A G → lim n A G /f n A G for each f ∈ I . We pick a Cauchy sequence { a n } in A G with a n +1 − a n ∈ f n A G for n ≥
1. Itsuffices to show that there is a ∈ A G such that a − a n ∈ f n A G for all n . Since f n A G ⊂ I n A ∩ A G It is automatically continuous in the I -adic topology. Since it the kernel of the continuous morphism A α − Id −−−→ Q g ∈ G A . UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 13 and A G is complete in the subspace topology, we see that there is an element b ∈ A G such that b − a n ∈ f n A G for all n ≥
1. We can replace each a n with b − a n to assume that a n ∈ f n A G .Finally, we show that a n ∈ f n A G for all n ≥
1. As then a = 0 clearly does the job. Ourassumption implies that a n = a n +1 + f n x n for some x n ∈ A G . We claim a n = f n ( x n + f x n +1 + f x n +2 + . . . ) . (3)The sum x n + f x n +1 + f x n +2 + . . . converges in A G as f m x n + m ∈ I m A ∩ A G ; so the right side ofthe equation (3) is well-defined. Also, f n ( x n + f x n +1 + f x n +2 + . . . )converges to a n because the partial sums equal a n − a n + m and a n + m ∈ I n + m A ∩ A G . This finishesthe proof. (cid:3) Lemma 3.2.2.
Let A be an admissible k + -algebra with a k + -action of a finite group G . Then(1) A G is complete in the ̟ -adic topology.(2) A G is saturated in A .(3) A is a finite as an A G -module. Proof.
The first claim is Lemma 3.2.1. The second claim is clear by k + -flatness of A . Thus we onlyneed to show the last claim.Lemma 2.1.10(1) guarantees that A G → A is integral. However, the proof of finiteness inLemma 2.1.10(3) is not applicable here since A is not necessarily finite type over k + : it is onlytopologically finite type.We now overcome this difficulty. Clearly, the morphism A G /̟A G → A/̟A is integral. But
A/̟A is a finite type k + /̟k + -algebra by our assumption, so A G /̟A G → A/̟A is a finite typemorphism. Since an integral map of finite type is finite, we conclude that a morphism A G /̟A G → A/̟A is finite. Therefore, the successive approximation argument (or [Sta19, Tag 031D]) impliesthat A is finite as an A G -module . (cid:3) Lemma 3.2.3 (Adic Artin-Tate) . Let A → B be a finite injective morphism of ̟ -adically complete k + -algebras. Suppose that B is a topologically finite type k + -algebra and A is a saturated submod-ule of B (in the sense of Definition 3.1.1). Then A is also a topologically finite type k + -algebra.The proof imitates the proof of Lemma 2.2.3; the main new difficulty is that we need to keeptrack of topological aspects of our algebras in order to work with topologically finite type algebrasin a meaningful way. Proof.
Since B is topologically finite type over k + , we can choose a finite set of elements x , . . . , x n such that the natural k + -linear continuous homomorphism p : k + h T , . . . , T n i → B that sends T i to x i is surjective.Since B is a finite A -module, we can choose some A -module generators y , . . . , y m ∈ B . Thechoice of x , . . . , x n and y , . . . , y m implies that there are some a i,j , a i,j,l ∈ A and relations x i = X j a i,j y i y i y j = X l a i,j,l y l . We consider the k + -algebra A ′ := k + h T i,j , T i,j,l i with a continuous k + -algebra homomorphism A ′ → A that sends T i,j to a i,j , and T i,j,l to a i,j,l . This map is well-defined as A is ̟ -adicallycomplete.By definition A ′ is topologically finite type over k + , and we claim that B is finite over A ′ sinceit is generated by y , . . . , y m as an A ′ -module. To see this we note that it suffices to show it mod ̟ by successive approximation (or [Sta19, Tag 031D]). However, it is easily seen to be finite mod ̟ due to the relations above.We use Lemma 3.1.6(4) to conclude that A is finite over A ′ as a saturated submodule of a finite A ′ -module B . This finishes the proof since a finite algebra over a topologically finite type k + -algebrais also topologically finite type. (cid:3) Corollary 3.2.4.
Let A be an admissible k + -algebra with a k + -action of a finite group G . Then A G is an admissible k + -algebra, the induced topology on A G coincides with the ̟ -adic topology,and A is a finitely presented A G -module. Proof.
We use Lemma 3.2.1 and Lemma 3.2.2 to say that A G is ̟ -adically complete, and A G → A issaturated. Then Lemma 3.2.3 guarantees that A G is a topologically finitely generated k + -algebra.Now A is a finite module over a topologically finitely generated k + -algebra A G , so the inducedtopology on A G coincides with the ̟ -adic topology by Lemma 3.1.6(5).Now Lemma 2.2.2 (1) implies that A G is k + -flat as it is torsion free. Therefore, Lemma 3.1.6 (3)guarantees that A is a finitely presented A G -module. (cid:3) Remark 3.2.5.
One can show that the ̟ -adic topology on A G coincides with the induced topologyfrom first principles. But we prefer the proof above as it generalizes better to the topologicallyuniversally adhesive situation (see Definition A.3.1).Namely, Lemma 3.2.3 and Corollary 3.2.4 hold over any I -adically complete base ring R thatis topologically universally adhesive (see Definition A.3.1). We refer to Lemma A.3.6 and Corol-lary A.3.7 for the precise results.Finally, we are ready to show that X /G is an affine admissible formal k + -scheme if X is so. Proposition 3.2.6.
Let X = Spf A be an affine admissible formal k + -scheme with a k + -action of afinite group G . Then the natural map φ : X /G → Y = Spf A G is a k + -isomorphism of topologicallylocally ringed spaces. In particular, X /G is an admissible formal k + -scheme. Proof. Step 0.
Spf A G is an admissible formal k + -scheme : The k + -algebra A is admissible byRemark 3.1.5(3) (and the analogous fact for topologically finitely generated morphisms). Now theclaim immediately follows from Corollary 3.2.4. Step 1. φ is a homeomorphism : This is completely analogous to Step 1 of Proposition 2.1.13.We only need to show that p : Spf A → Spf A G is a surjective, finite morphism with fibers beingexactly G -orbits.Lemma 3.2.2 says that Spf A → Spf A G is finite. We note that surjectivity of Spec A → Spec A G obtained in Lemma 2.1.10(2) implies that any prime ideal p of A G lifts to a prime ideal P in A . If p is open (i.e. it contains ̟ n for some n ), then so is P . Therefore, the morphism Spf A → Spf A G is surjective.Now we note that a prime ideal P ⊂ A is open if and only if so is g ( P ) for g ∈ G . SoLemma 2.1.10(2) ensures that the fibers of Spf A → Spf A G are exactly G -orbits. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 15
Step 2. φ is an isomorphism of topologically locally ringed spaces: We already know that φ is ahomeomorphism. So the only thing that we need to show here is that the morphism O Y → φ ∗ O X /G is an isomorphism of topological sheaves. Using the basis of basic affine opens in Y , it suffices toshow that (cid:0) A G (cid:1) { f } → (cid:0) A { f } (cid:1) G (4)is a topological isomorphism. Corollary 3.2.4 ensures that both sides have the ̟ -adic topology, sowe can ignore the topologies.Now we show that (4) is an (algebraic) isomorphism. We note that A { f } ≃ (cid:0) A G (cid:1) { f } b ⊗ A G A ≃ (cid:0) A G (cid:1) { f } ⊗ A G A, where the second isomorphism follows from Lemma 3.1.6(1) and finiteness of A over A G . Therefore,it suffices to show that the natural morphism (cid:0) A G (cid:1) { f } → (cid:16)(cid:0) A G (cid:1) { f } ⊗ A G A (cid:17) G is an isomorphism of k + -algebras. This follows from Lemma 2.1.12 and Lemma 3.1.6(6). (cid:3) Remark 3.2.7.
Proposition 3.2.6 can be generalized to the case of an affine, universally adhesivebase S = Spf R (see Definition A.3.9). We refer to Proposition A.3.8 for the precise statement.3.3. General Case.
The main goal of this section is to globalize the results of the previous section.This is very close to what we did in the schematic situation in the proof of Theorem 2.1.16.
Lemma 3.3.1.
Let X be a formal S -scheme with an S -action of a finite group G . Suppose thateach point x ∈ X admits an open affine subscheme V x that contains the orbit G.x . Then the sameholds with X replaced by any G -stable open formal subscheme U ⊂ X . Proof.
This follows follows easily from Lemma 2.1.7 as | X | ≃ | X × Spf k + Spec k + /̟ | and | S | = | Spf k + | ≃ | Spec k + /̟ | . Thus, we can reduce the statement to the case of schemes. (cid:3)
Lemma 3.3.2.
Let X be a formal S -scheme with an S -action of a finite group G . Suppose thatfor any point x ∈ X there is an open affine subscheme V x that contains the orbit G.x . Then eachpoint x ∈ X has a G -stable open affine neighborhood U x ⊂ X . Proof.
Again, this easily follows from Lemma 2.1.14 as an open subscheme U ⊂ X is affine if andonly if U := U × Spf k + Spec k + /̟ is affine [FK18, Proposition I.4.1.12]. (cid:3) Remark 3.3.3.
We note that the condition of Lemma 3.3.2 is automatically fullfiled if the specialfiber X := X × Spf k + Spec k + / m k is quasi-projective over Spec k + / m k . This follows easily fromProposition 2.1.15.Now we are ready to formulate and prove the main result of this section. Theorem 3.3.4.
Let X be an admissible formal k + -scheme with a k + -action of a finite group G .Suppose that each point x ∈ X admits an affine neighborhood V x containing G.x . Then X /G is anadmissible formal k + -scheme. Moreover, it satisfies the following properties:(1) π : X → X /G is universal in the category of G -invariant morphisms to topologically locallyringed spaces over S . (2) π : X → X /G is a surjective, finite, topologically finitely presented morphism (in particular,it is closed).(3) Fibers of π are exactly the G -orbits.(4) The formation of the geometric quotient commutes with flat base change, i.e. for any flat,topologically finite type k + -morphism Z → X /G , the geometric quotient ( X × X /G Z ) /G is an admissible formal k + -schemes, and the natural morphism ( X × X /G Z ) /G → Z is anisomorphism. Proof. Step 1. The geometric quotient X /G is an admissible formal k + -scheme : The same proofas used in the proof of Theorem 2.1.16 just goes through. We firstly reduce to the case of an affine X = Spf A by choosing a G -stable open affine covering, and then use Proposition 3.2.6 to show theclaim in the affine case. Step 2. π : X → X /G is surjective, finite, topologically finitely presented, and fibers are exactlythe G -orbits : The morphism is clearly surjecitve with fibers being exactly the G -orbits.To show that it is finite and topologically finitely presented, we can assume that X = Spf A is affine. Lemma 3.2.2 says that X → X /G is finite. Corollary 3.2.4 ensures that A is finitelypresented as an A G -module. Therefore, it is topologically finitely presented as an A G -algebrabecause [Bos14, Proposition 7.3/10] gives that A G → A is topologically finitely presented if andonly if A G /̟ n A G → A/̟ n A is finitely presented for any n ≥ Step 3. π : X → X /G is universal and commutes with flat base change : The universality isessentially trivial (see Remark 3.1.8). To show the latter claim, we can again assume that X = Spf A and Z = Spf B are affine. Then the claim boils down to showing that the natural map B → ( A b ⊗ A G B ) G is a topological isomorphism. Now we note that Lemma 2.2.2(1) implies that A ⊗ A G B is already ̟ -adically complete as it is a finite module over a topologically finite type k + -algebra B . Therefore,it suffices to show that the natural map B → ( A ⊗ A G B ) G (5)is a topological isomorphism. Both sides have the ̟ -adic topology by Corollary 3.2.4. So we canignore the topologies. Now (5) is an isomorphism by Lemma 2.1.12 and flatness of A G → B (seeRemark 3.1.5). (cid:3) Remark 3.3.5.
Theorem 3.3.4 can be generalized to the case of a locally universally adhesive base S (see Definition A.3.9). We refer to Theorem A.3.15 for the precise statement.3.4. Comparison between the schematic and formal quotients.
For this section, we fix amicrobial valuation ring k + (see Definition 3.1.1) with a choice of a pseudo-uniformizer ̟ .If X is a flat, locally finite type k + -scheme, we define b X to be the formal ̟ -adic completionof X . This is easily seen to be an admissible formal b k + -scheme with a b k + -action of G . Using theuniversal property of geometric quotients, there is a natural morphism b X/G → [ X/G . Theorem 3.4.1.
Let X be a flat, locally finite type k + -scheme with a k + -action of a finite group G . Suppose that any orbit G.x ⊂ X lies in an affine open subset V x . The same holds for its ̟ -adiccompletion b X with the induced b k + -action of G , and the natural morphism: b X/G → [ X/G is an isomorphism.
UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 17
Proof. Step 1. The condition of Theorem 3.3.4 is satisfied for b X with the induced action of G :Firstly, we observe that b X is b k + -admissible by the discussion above. Now Lemma 2.1.14 saysthat our assumption on X implies that there is a covering of X = ∪ i ∈ I U i by affine, open G -stablesubschemes. Then b X = ∪ i ∈ I b U i is an open covering of b X by affine, G -stable open formal subschemes.In particular, every orbit lies in an open, affine open formal subscheme of b X . Step 2. We show that b X/G → [ X/G is an isomorphism : We have a commutative diagram b X b X/G [ X/G. π b X c π X φ of admissible formal b k + -schemes. We want to show that φ is an isomorphism. To prove the claim,we can assume that X = Spec A is affine by passing to an open covering of X by G -stable affines.Then X/G ≃ Spec A G , b X/G ≃ Spf b A G , and φ can be identified with the mapSpf( b A ) G → Spf [ ( A G )induced by the continuous homomorphism c A G → ( b A ) G (6)whose target has the ̟ -adic topology by Corollary 3.2.4. So it suffices to show that this map is atopological isomorphism for any flat, finitely generated k + -flat algebra A . Both sides have ̟ -adictopology, so we can ignore the topologies.We note that Corollary 2.2.4 shows that A G is a finite type k + -algebra and A is a finite A G -module, so [Bos14, Lemma 7.3/14] gives that the natural homomorphism A ⊗ A G c A G → b A is an (algebraic) isomorphism. Thus we can identify (6) with the natural homomorphism c A G → (cid:16) A ⊗ A G c A G (cid:17) G that is an (algebraic) isomorphism by Lemma 2.1.12 and flatness of the map A G → c A G (see [Bos14,Lemma 8.2/2]). (cid:3) Remark 3.4.2.
Theorem 3.4.1 has a version over any topologically universally adhesive base(
R, I ) (see Definition A.3.1). We refer to Theorem A.3.16 for the precise statement.4. Quotients of Strongly Noetherian Adic Spaces
We discuss the existence of quotient of some class of analytic adic spaces by an action of a finitegroup G . We refer the reader to Appendix B for the review of the main definitions and facts fromthe theory of Huber rings and corresponding adic spaces.The strategy to construct quotients is close to the one used in Section 2 and Section 3. We firstlyconstruct the candidate space X/G that is, a priori, only a valuation space. This constructionclearly satisfies the universal property, but it is not clear if
X/G is an adic space. We resolve thisissue by firstly showing that it is an adic space if X is affinoid. Then we argue by gluing to provethe claim for a larger class of adic spaces. We do not assume that R is I -adically complete. We point out the two main complications compared to Section 3 (and Section 2). The first newissue that is not seen in the world of formal schemes is that the notion of a finite (resp. topologicallyfinite type) morphism of Huber pairs (
A, A + ) → ( B, B + ) is more involved since there is an extracondition on the morphism A + → B + that makes the theory more subtle (see Definition B.2.1 andDefinition B.2.6).The second issue is that the underlying topological space Spa( A, A + ) of a Huber pair ( A, A + )is more difficult to express in terms of the pair ( A, A + ). It is the set of all valuations on A withcorresponding continuity and integrality conditions. So one needs some extra work to identifySpa( A G , A + ,G ) with Spa( A, A + ) /G even in the affine case.4.1. The Candidate Space
X/G . For the rest of the section we fix a locally strongly noetheriananalytic adic space S (see Definition B.2.15). Example 4.1.1.
An example of a strongly noetherian Tate affinoid adic space S is Spa( k, k + ) fora microbial valuation ring k + . Definition 4.1.2.
Let G be a finite group, and X a valuation locally topologically ringed spaceover S with a right S -action of G . The geometric quotient X/G = ( | X/G | , O X/G , { v x } x ∈ X/G , h )consists of: • the topological space | X/G | := | X | /G with the quotient topology. We denote by π : | X | →| X/G | the natural projection, • the sheaf of topological rings O X/G := ( π ∗ O X ) G with the subspace topology, • for any x ∈ X/G , the valuation v x defined as the composition of the natural morphism k ( x ) → k ( x ) and the valuation v x : k ( x ) → Γ v x ∪ { } , where x ∈ p − ( x ) is any lift of x . • the morphism h : X/G → S defined by the pair ( h, h ), where h : | X | /G → S is the uniquemorphism induced by f : X → S and h is the natural morphism O S → h ∗ (cid:0) O X/G (cid:1) = h ∗ (cid:16) ( π ∗ O X ) G (cid:17) = ( h ∗ ( π ∗ O X )) G = ( f ∗ O X ) G that comes from G -invariance of f . Remark 4.1.3.
We note that Lemma 2.1.3 ensures that
X/G is a valuative topologically locallyringed S -space, and π : X → X/G is a morphism of valuative topologically locally ringed S -spaces(so X/G → S is too). It is trivial to see that the pair ( X/G, π ) is a universal object in the categoryof G -invariant morphisms to valuative topologically locally ringed S -spaces.Our main goal is to show that under some assumptions, X/G is a locally topologically finitetype adic S -space when X is. We start with the case of affinoid adic spaces and then move to thegeneral case.4.2. Affinoid Case.
For the rest of this section, we assume that S = Spa( R, R + ) is a stronglynoetherian Tate affinoid.We show that X/G is a topologically finite type adic S -space when X = Spa( A, A + ) for atopologically finite type complete ( R, R + )-Tate-Huber pair ( A, A + ).We start the section by discussing algebraic properties of the Tate-Huber pair (cid:0) A G , A + G (cid:1) . Inparticular, we show that it is topologically of finite type over ( R, R + ). The main new input isthe “analytic” Artin-Tate Lemma 4.2.4. Then we show that the canonical morphism X/G → Lemma 2.1.3 ensures that ( | X/G | , O X/G ) is a locally ringed space, so k ( x ) is well-defined. One can show that v x is independent of a choice of x similarly to Lemma 2.1.3. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 19
Spa( A G , A + ,G ) is an isomorphism. In particular, X/G is an adic space, topologically finite typeover S . Lemma 4.2.1.
Let (
A, A + ) be a complete ( R, R + )-Tate-Huber pair with an ( R, R + )-action of afinite group G . Then(1) A has a G -stable pair of definition ( A , ̟ ) such that A ⊂ A + .(2) The subspace topology on ( A G , ̟ ) coincides with the ̟ -adic topology.(3) ( A G , ̟ ) is a complete pair of definition of A G with the subspace topology. In particular, A G is a Huber ring.(4) ( A G , A + ,G ) with the subspace topology is a Tate-Huber pair. Proof.
We note that A is Tate since R is. We choose a pair of definition ( R , ̟ ) of R and acompatible pair of definition ( A ′ , ̟ ) of A . Then [Hub93a, Proposition 1.1] ensures that a subring A ′ ⊂ A is a ring of definition if and only if A ′ is open and bounded. So we can replace A ′ with A ′ ∩ A + and ̟ with a power to assume that A ′ ⊂ A + .Now loc.cit. implies that ( A , ̟ ) := \ g ∈ G g (cid:0) A ′ (cid:1) , ̟ is a pair of definition in A contained in A + , and it is G -stable by construction.To show that the subspace topology in A G coincides with the ̟ -adic topology, it suffices to showthat ̟ n A ∩ A G = ̟ n A G . This can be easily seen from the fact that ̟ is a unit in A (and so anon zero divisor in A ).Now we note that A G is complete in the subspace topology since the action of G on A is clearlycontinuous. Therefore, it is complete in the ̟ -adic topology as these topologies were shown to beequivalent. Also, we note that A G with the subspace topology is clearly open and bounded in A G ,so it is a ring of definition by [Hub93a, Proposition 1.1].Finally, we note that clearly A + ,G ⊂ A ◦ ∩ A G ⊂ (cid:0) A G (cid:1) ◦ is open and integrally closed subring of (cid:0) A G (cid:1) ◦ . So ( A G , A + ,G ) is a Tate-Huber pair. (cid:3) Corollary 4.2.2.
Let (
A, A + ) be a complete ( R, R + )-Tate-Huber pair with an ( R, R + )-action ofa finite group G . Then the action of G on A is continuous. Proof.
We choose a G -stable pair of definition ( A , ̟ ) as in Lemma 4.2.1(1). Then it sufficesto show that the action of G on A is continuous. This is clear because A carries the ̟ -adictopology. (cid:3) Lemma 4.2.3.
Let (
A, A + ) be a topologically finite type (see Definition B.2.1) complete ( R, R + )-Tate-Huber pair with an ( R, R + )-action of a finite group G . Then the morphism ( A G , A + ,G ) → ( A, A + ) is a finite morphism of complete Huber pairs (see Definition B.2.6). Proof.
Lemma 2.1.10 gives that the morphisms A G → A and A + ,G → A + are integral. So weonly need to show that A is module-finite over A G . Lemma 4.2.1 and Lemma B.2.4 (applied to( R, R + ) → ( A G , A + ,G ) → ( A, A + )) ensure that ( A G , A + ,G ) → ( A, A + ) is a topologically finite typemorphism of complete Tate-Huber pairs with A + ,G → A + being integral. Therefore, Lemma B.2.9implies that ( A G , A + ,G ) → ( A, A + ) is finite. (cid:3) We abuse the notation and consider ̟ as an object of A via the natural morphism R → A . Lemma 4.2.4 (Analytic Artin-Tate) . Let i : ( A, A + ) → ( B, B + ) be a finite injective morphismof complete Tate-Huber ( R, R + )-pairs. If ( B, B + ) is topologically finite type ( R, R + )-Tate-Huberpair, then so is ( A, A + ).The proof of Lemma 4.2.4 imitates the proof of the Adic Artin-Tate Lemma (Lemma 3.2.3), butit is more difficult due to the issue that we need to control the integral aspect of Definition B.2.6.We recommend the reader to look at the proof of Lemma 3.2.3 before reading this proof. Proof. Step 0. Preparation for the proof : We choose a pseudo-uniformizer ̟ ∈ R and an open,surjective morphism f : R h X , . . . , X n i ։ B such that B + is integral over f ( R + h X , . . . , X n i ). We denote by x i ∈ B + the image f ( X i ). Step 1. We choose “good” A -module generators y , . . . , y m of B : Remark B.2.10 implies thatthere is a compatible choice of rings of definitions A ⊂ A , B ′ ⊂ B containing all x i such that B ′ is a finite A -module. Then we choose A -module generators y , . . . , y m of B ′ . Since B ≃ B ′ (cid:2) ̟ (cid:3) , A ≃ A (cid:2) ̟ (cid:3) , we conclude that y , . . . , y m are also A -module generators of B . The crucial propertyof this choice of A -module generators is that there exist a i,j , a i,j,k ∈ A ⊂ A + such that x i = X j a i,j y j ,y i y j = X k a i,j,k y k . Step 2. We define another ring of definition B : We consider the unique surjective, continuous R -algebra homomorphism g : R h X , . . . , X n , Y , . . . , Y m , T i,j , T i,j,k i → B defined by g ( X i ) = x i , g ( Y j ) = y j , g ( T i,j ) = a i,j , and g ( T i,j,k ) = a i,j,k . This morphism is automati-cally open by Remark B.2.3.We define B := g ( R h X , . . . , X n , Y , . . . , Y m , T i,j , T i,j,k i ), where R is a ring of definition in R compatible with A (see [Hub93a, Corollary 1.3(ii)] for its existence). This is clearly an open andbounded subring of B , so it is a ring of definition.By construction, B contains f ( R + h X , . . . , X n i ), and B /̟B is generated as an R /̟R -algebra by the classes x i , y j , a i,j , and a i,j,k . Step 3. We show that B + is integral over R + B : We note that B + is integral over f ( R + h X , . . . , X n i ) = f ( R + R h X , . . . , X n i ) = R + f ( R h X , . . . , X n i ) . Therefore, it is integral over R + B since it contains R + f ( R h X , . . . , X n i ) by the previous Step. Step 4. We show that ( B, B + ) is finite over ( R h T i,j , T i,j,k i , R + h T i,j , T i,j,k i ): We recall that a i,j , a i,j,k ∈ A ⊂ A + for all i , j , k . So, we can use the universal property of restricted power series todefine a continuous morphism of complete Tate-Huber pairs: r : (cid:0) R h T i,j , T i,j,k i , R + h T i,j , T i,j,k i (cid:1) → ( A, A + )as the unique continuous R -algebra morphism such that r ( T i,j ) = a i,j , r ( T i,j,k ) = a i,j,k . UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 21
We also define the morphism t : (cid:0) R h T i,j , T i,j,k i , R + h T i,j , T i,j,k i (cid:1) → ( B, B + )as the composition of r and i .We now show that B is finite over R h T i,j , T i,j,k i . Note that this actually makes sense since thenatural morphism R h T i,j , T i,j,k i → B factors through B by the choice of B . We consider the reduction B /̟B and claim that it isfinite over R h T i,j , T i,j,k i /̟ = ( R /̟ ) [ T i,j , T i,j,k ] . Indeed, we know that B /̟B is generated as an R /̟R -algebra by the elements x , . . . , x n , y , . . . , y m , a i,j , a i,j,k . However, we note that a i,j = t ( T i,j ) and a i,j,k = t ( T i,j,k ). Thus, we can conclude that B /̟ isgenerated as an R /̟ [ T i,j , T i,j,k ]-algebra by the elements x , . . . , x n , y , . . . , y m . Recall that the choice of x i and y j implies that each of x i is a linear combination of y j with coeffi-cients in a i,j = t ( T i,j ). This implies that B /̟B is generated as an ( R /̟R )[ T i,j , T i,j,k ]-algebraby y , . . . , y m . But again, the same argument shows that each product y i y j can be expressed as alinear combination of y k with coefficients a i,j,k = t ( T i,j,k ). This implies that y , . . . , y m are actu-ally ( R /̟R )[ T i,j , T i,j,k ]-module generators for B /̟B . Now we use a successive approximationargument (or [Sta19, Tag 031D]) to conclude that B is finite over R h T i,j , T i,j,k i .We conclude that B is a finite module over R h T i,j , T i,j,k i since B = B (cid:20) ̟ (cid:21) , and R h T i,j , T i,j,k i = R h T i,j , T i,j,k i (cid:20) ̟ (cid:21) . Thus, we are only left to show that B + is integral over R + h T i,j , T i,j,k i . Step 3 implies that B + is integral over B R + , so it suffices to show that B R + is integral over R + h T i,j , T i,j,k i . But thiseasily follows from the fact that B is finite over R h T i,j , T i,j,k i . Step 5. We show that ( A, A + ) is finite over ( R h T i,j , T i,j,k i , R + h T i,j , T i,j,k i ) : Note that R h T i,j , T i,j,k i is noetherian since R is strongly noetherian by the assumption. Therefore, we see that A must bea finite R h T i,j , T i,j,k i -module as a submodule of a finite module B . Moreover, we see that A + isequal to the intersection B + ∩ A because ( B, B + ) is a finite ( A, A + )-Tate-Huber pair. This impliesthat A + is integral over the image r ( R + h T i,j , T i,j,k i ). We conclude that the complete Huber pair( A, A + ) is finite over ( R h T i,j , T i,j,k i , R + h T i,j , T i,j,k i ). Therefore, it is topologically finite type over( R, R + ) by Lemma B.2.8 and Lemma B.2.4. (cid:3) Corollary 4.2.5.
Let (
A, A + ) be a topologically finite type complete ( R, R + )-Tate-Huber pairwith an ( R, R + )-action of a finite group G . Then the complete Tate-Huber pair (cid:0) A G , A + ,G (cid:1) istopologically finite type over ( R, R + ), and the natural morphism (cid:0) A G , A + ,G (cid:1) → ( A, A + ) is a finitemorphism of complete Tate-Huber pairs. Proof.
Lemma 4.2.3 gives that (cid:0) A G , A + ,G (cid:1) → ( A, A + ) is a finite morphism of complete Tate-Huberpairs. So Lemma 4.2.4 guarantees that (cid:0) A G , A + G (cid:1) is a topologically finite type complete ( R, R + )-Tate-Huber pair. (cid:3) Theorem 4.2.6.
Let X = Spa( A, A + ) be a topologically finite type affinoid adic S = ( R, R + )-spacewith an S -action of a finite group G . Then the natural morphism φ : X/G → Y = Spa (cid:0) A G , A + G (cid:1) is an isomorphism over S . In particular, X/G is topologically finite type affinoid adic S -space.We adapt the proofs of Proposition 2.1.13 and 3.2.6. However, there are certain complicationsdue to the presence of higher rank points. Namely, there are usually many different points v ∈ Spa (
A, A + ) with the same support p . Thus in order to study fibers of the map X → Y we need towork harder than in algebraic and formal setups. Proof. Step 0. Preparation : The S -action of G on Spa( A, A + ) induces an ( R, R + )-action of G on( A, A + ). By Corollary 4.2.5, ( A G , A + ,G ) is topologically finite type over ( R, R + ). In particular, Y = Spa( A G , A + ,G ) is an adic space , and it is topologically finite type over S .Now we recall that there is a natural map of valuation spaces p ′ : Spv A → Spv A G , where Spv A (resp. Spv A G ) is the set of all valuations on the ring A (resp. A G ). We have the commutativediagram Spa ( A, A + ) Spa (cid:0) A G , A + G (cid:1) Spv A Spv A G Spec A Spec A Gpp ′ p ′′ with the upper vertical maps being the forgetful maps and the lower vertical maps being the mapsthat send a valuation to its support. Step 1. The natural map p ′ : Spv A → Spv A G is surjective and G acts transitively on fibers: Recall that data of a valuation v ∈ Spv A is the same as data of a prime ideal p v ⊂ A (its support)and a valuation ring R v ⊂ k ( p ).To show surjectivity of p ′ , pick any valuation v ∈ Spv A G ; we want to lift it to a valuation of A .We use Lemma 2.1.10 to find a prime ideal q ⊂ A that lifts the support p v := supp( v ) ⊂ A G , so k ( q ) is finite over k ( p v ) since A is A G -finite.Now we use [Mat86, Theorem 10.2] to dominate a valuation ring R v ⊂ k ( p v ) by some valuationring R w ⊂ k ( q ). This provides us with a valuation w : A → Γ w ∪{ } such that p ′ ( w ) = v . Therefore,the map p ′ is surjective.As for the transitivity of the G -action, we note that Lemma 2.1.10 implies that G acts transitivelyon the fiber ( p ′′ ) − ( p v ). Furthermore, [Bou98, Ch.5, §
2, n.2, Theorem 2] guarantees that, for anyprime ideal q ∈ ( p ′′ ) − ( p v ), the stabilizer subgroup G q := Stab G ( q )surjects onto the automorphism group Aut( k ( q ) /k ( p v )). We use [Bou98, Ch. 6, §
8, n.6, Proposition6] to see that there is a bijection between the sets (cid:26)
Valuations w on k ( q )restricting to v on k ( p v ) (cid:27) ↔ (cid:8) Maximal ideals in Nr k ( q ) ( R v ) (cid:9) , The structure presheaf O Y is a sheaf on Y by [Hub94, Theorem 2.5]. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 23 where Nr k ( q ) ( R v ) is the integral closure of R v in the field k ( q ). Now we use [Mat86, Theorem 9.3(iii)]to conclude that Aut( k ( q ) /k ( p v )) (and therefore G q ) acts transitively on the set of maximal idealsof Nr k ( q ) ( R v ). As a consequence, G q acts transitively on the set of valuations w ∈ p ′− ( v ) with thesupport q . Therefore, G acts transitively on p ′− ( v ) for any v ∈ Spv A G . Step 2. We show that p : Spa( A, A + ) → Spa( A G , A + ,G ) is surjective, and G acts transitivelyon fibers : We recall that Spa( A, A G ) (resp. Spa( A G , A + ,G )) is naturally a subset of Spv( A ) (resp.Spv( A G )). Therefore, it suffices (by Step 1) to show that, for any v ∈ Spa( A G , A + ,G ), any w ∈ p ′− ( v ) is continuous and w ( A + ) ≤ w ( A + ) ≤ A + is integral over A + ,G . So we only need to show that thevaluation w ∈ Spv( A ) is continuous. Lemma 4.2.7.
Let A be a Tate ring with a pair of definition ( A , ̟ ), where ̟ is a pseudo-uniformizer. Then a valuation v : A → Γ v ∪ { } is continuous if and only if: • The value v ( ̟ ) is cofinal in Γ v , • v ( a̟ ) < v for any a ∈ A . Proof. [Sem15, Corollary 9.3.3] (cid:3)
We choose a G -stable pair of definition ( A , ̟ ) from Lemma 4.2.1. Then [Bou98, Ch. 6, §
8, n.1,Corollary 1] gives that Γ w / Γ v is a torsion group. Therefore w ( ̟ ) = v ( ̟ ) is cofinal in Γ w if it iscofinal in Γ v . In particular, w ( ̟ ) < w ( A + ) ≤ v | A + ,G = w | A + ,G , w ( a̟ ) = w ( a ) w ( ̟ ) < w ( a ) ≤ a ∈ A ⊂ A + . Step 3. We show that φ : X/G → Y is a homeomorphism: Step 2 implies that φ is a bijection.Now note that that p : X → Y is a finite, surjective morphism of strongly noetherian adic spaces.Therefore, it is closed by [Hub96, Lemma 1.4.5(ii)]. In particular, it is a topological quotient mor-phism. The map π : X → X/G is a topological quotient morphism by its construction. Hence, φ isa homeomorphism. Step 4. We show that φ is an isomorphism of valuative topologically locally ringed spaces : Firstly,Remark B.1.2 implies that it suffices to show that the natural morphism O Y → φ ∗ O X/G is an isomorphism of sheaves of topological rings. Using the basis of rational subdomains in Y , itsuffices to show that A G (cid:28) f s , . . . f n s (cid:29) → (cid:18) A (cid:28) f s , . . . , f n s (cid:29)(cid:19) G (7)is a topological isomorphism for any f , . . . , f n generating the unit ideal in A G . Lemma 4.2.3 givesthat (7) is a continuous morphism of complete Tate rings. So the Banach Open Mapping Theorem[Hub94, Lemma 2.4 (i)] guarantees that it is automatically open (and so a homeomorphism) if itis surjective. Thus, we can ignore the topologies.Now we show that (7) is an (algebraic) isomorphism. We note that A ⊗ A G A G (cid:28) f s , . . . , f n s (cid:29) ≃ A (cid:28) f s , . . . , f n s (cid:29) , by Corollary B.3.7. Therefore, it suffices to show that A G (cid:28) f s , . . . f n s (cid:29) → (cid:18) A ⊗ A G A G (cid:28) f s , . . . , f n s (cid:29)(cid:19) G is an isomorphism. This follows from Lemma 2.1.12 and flatness of the map A G → A G D f s , . . . , f n s E obtained in [Hub94, Case II.1.(iv) on p. 530]. (cid:3) General Case.
The main goal of this section is to globalize the results of the previous section.This is very close to what we did in the formal situation in the proof of Theorem 3.3.4. The mainissue is that the adic analog of Lemma 3.3.2 is more difficult to show.For the rest of the section, we fix a locally strongly noetherian analytic adic space S (see Defi-nition B.2.15). Lemma 4.3.1.
Let X = Spa( A, A + ) be a pre-adic Tate affinoid , and V ⊂ X an open pre-adicsubspace. Then any finite set of points F ⊂ V is contained in an affinoid pre-adic subspace of V .Our proof uses an adic analog of the theory of “formal” models of rigid spaces in an essentialway. It might be possible to justify this claim directly from the first principles, but it seems quitedifficult due to the fact that the complement X \ V does not have a natural structure of a pre-adicspace.In what follows, for any topological space Z with a map Z → Spec A + , we denote by Z the fiberproduct Z × Spec A + Spec A + /̟ in the category of topological spaces. Proof.
First of all, we note that rational subdomains form a basis in Spa(
A, A + ), and they arequasi-compact. Therefore, we can find a quasi-compact open subspace F ⊂ V ′ ⊂ V , so we mayand do assume that V is quasi-compact.We consider the affine open immersion U = Spec A → S = Spec A + . And define the category of U -admissible modifications Adm
U,S to be the category of projectivemorphisms f : Y → S that are isomorphisms over U . Then [Bha, Theorem 8.1.2 and Remark8.1.8] shows that X := (cid:18) lim f : Y → Spec A + | f ∈ Adm
U,S Y (cid:19) × Spec A + Spec A + /̟ ≃ lim f ∈ Adm
U,S Y admits a canonical morphism X → Spa(
A, A + ) that is a homeomorphism. Since V ⊂ X is quasi-compact, [Sta19, Tag 0A2P] implies that there is a U -admissible modification Y → Spec A + and aquasi-compact open V ′ ⊂ Y such that π − ( V ′ ) = V for the projection map π : X → Y .Now V ′ is a quasi-projective scheme over Spec A + /̟ . Therefore, [EGA II, Corollaire 4.5.4]implies that there is an open affine subscheme W ⊂ V ′ containing π ( F ). Therefore, π − ( W ) ⊂ Spa(
A, A + ) contains F , and (the proof of) [Bha, Corollary 8.1.7] implies that π − ( W ) is affinoid . (cid:3) We do not assume that the structure presheaf O X is a sheaf. We emphasize that a projective morphism is not required to be finitely presented The inverse limit giving the preimage in the statement is shown to be affinoid in the proof.
UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 25
Lemma 4.3.2.
Let X be a pre-adic space with an action of a finite group G . Suppose that eachpoint x ∈ X admits an open affinoid pre-adic subspace V x that contains the orbit G.x . Then thesame holds with X replaced by any G -stable open pre-adic subspace U ⊂ X . Proof.
The proof is analogous to that of Lemma 2.1.7. One only needs to use Lemma 4.3.1 in placeof [EGA II, Corollaire 4.5.4]. (cid:3)
Lemma 4.3.3.
Let X be a locally topologically finite type adic S -space with an S -action of a finitegroup G . Suppose that for any point x ∈ X there is an open affinoid adic subspace V x ⊂ X thatcontains the orbit G.x . Then each point x ∈ X has a G -stable strongly noetherian Tate affinoidneighborhood U x ⊂ X . Proof.
The proof is similar to that of Lemma 2.1.7. Lemma 4.3.2 allows to reduce to the case S a strongly noetherian Tate affinoid space. Then for a separated X , U x := T g ∈ G g ( V x ) is astrongly noetherian Tate affinoid (see Corollary B.7.5) and does the job. In general, Lemma 4.3.2guarantees that one can replace X with the separated open adic subspace T g ∈ G g ( V x ) and reduceto the separated case. (cid:3) Theorem 4.3.4.
Let X be a locally topologically finite type adic S -space with an S -action of afinite group G . Suppose that each point x ∈ X admits affinoid open neighborhood V x containing G.x . Then
X/G is a locally topologically finite type adic S -space. Moreover, it satisfies the followingproperties:(1) π : X → X/G is universal in the category of G -invariant morphisms to valuative topologi-cally locally ringed S -spaces.(2) π : X → X/G is a finite, surjective morphism (in particular, it is closed).(3) Fibers of π are exactly the G -orbits.(4) The formation of the geometric quotient commutes with flat base change, i.e. for any flatmorphism Z → X/G (see Definition B.4.1) of locally strongly noetherian analytic adicspaces, the geometric quotient ( X × X/G Z ) /G is an adic space, and the natural morphism( X × X/G Z ) /G → Z is an isomorphism. Proof. Step 1.
X/G is a topologically locally finite type adic S -space : Similarly to Step 1 of Theo-rem 2.1.16, we can use Lemma 4.3.2 and Lemma 4.3.3 to reduce to the case of a strongly noetherianTate affinoid S = Spa( R, R + ) and affinoid X = Spa( A, A + ). Then the claim follows from Theo-rem 4.2.6. Step 2. π : X → X/G is surjective, finite, and fibers are exactly the G -orbits : Similarly to Step 1,we can assume that X and S are affinoid. Then it follows from Lemma 4.2.3 and Theorem 4.2.6. Step 3. π : X → X/G is universal and commutes with flat base change : The universality isessentially trivial (Remark 2.1.5). To show the latter claim, we can again assume that X =Spa( A, A + ) and S = Spa( R, R + ) are strongly noetherian Tate affinoids and it suffices to considerstrongly noetherian Tate affinoid Z = Spa( B, B + ). Then the construction of the quotient impliesthat it suffices to show that the natural morphism of Tate-Huber pairs( B, B + ) → (cid:16) ( B, B + ) b ⊗ ( A G ,A + ,G ) ( A, A + ) (cid:17) G =: ( C, C + ) (8)is a topological isomorphism.We can ignore the topologies to show that B → B b ⊗ A G A is a topological isomorphism since itssurjectivity would imply openness by the Banach Open Mapping Theorem [Hub94, Lemma 2.4 (i)]. Now Corollary 4.2.5 and Lemma B.3.6 ensure that B b ⊗ A G A ≃ B ⊗ A G A . Therefore, it suffices toshow that the natural map B → ( B ⊗ A G A ) G is an (algebraic) isomorphism. This follows from Lemma 2.1.12 and flatness of A → B justified inLemma B.4.3. (cid:3) Comparison of adic quotients and formal quotients.
For this section, we fix a complete,microbial valuation ring k + (see Definition 3.1.1) with fraction field k , and a choice of a pseudo-uniformizer ̟ .We consider the functor of adic generic fiber:( − ) k : (cid:26) admissible formal k + -schemes (cid:27) → (cid:26) Adic Spaces locally of topologicallyfinite type over Spa( k, k + ) (cid:27) that is defined in [Hub96, § d there). Given an affine admissible formal k + -scheme Spf( A ), this functor assigns the affinoid adic space Spa( A (cid:2) ̟ (cid:3) , A + ) where A + is the integralclosure of A in A (cid:2) ̟ (cid:3) .Let X be an admissible formal k + -scheme with a k + -action of a finite group G . Then X k isa locally topologically finite type adic Spa( k, k + )-space with a Spa( k, k + )-action of G . Using theuniversal property of geometric quotients, there is a natural morphism X k /G → ( X /G ) k . Theorem 4.4.1.
Let X be an admissible formal k + -scheme with a k + -action of a finite group G .Suppose that any orbit G.x ⊂ X lies in an affine open subset. Then the adic generic fiber X k with the induced Spa( k, k + )-action of G satisfies the assumption of Theorem 4.3.4, and the naturalmorphism: X k /G → ( X /G ) k is an isomorphism. Proof.
Similarly to Step 1 in the proof of Theorem 3.4.1, the condition of Theorem 4.3.4 is satisfiedfor X k with the induced action of G .To show that X k /G → ( X /G ) k is an isomorphism, similarly to Step 2 in the proof of Theo-rem 3.4.1 we can assume that X = Spf A is affine. Then we have to show that the natural map (cid:18) A G (cid:20) ̟ (cid:21) , ( A G ) + (cid:19) → A (cid:20) ̟ (cid:21) G , ( A + ) G ! is an isomorphism of Tate-Huber pairs.Lemma 2.1.12 implies that the map A (cid:2) ̟ (cid:3) G → A G (cid:2) ̟ (cid:3) is an algebraic isomorphism. This is atopological isomorphism since both sides have A G as a ring of definition (see Corollary 3.2.4 andLemma 4.2.1). Therefore, we are only left to show that the natural map ( A G ) + → ( A + ) G is an(algebraic) isomorphism.Clearly, A + is integral over A , and so it is integral over A G by Lemma 2.1.10(1). Hence, ( A + ) G is integral over ( A G ) + . Since ( A G ) + is integrally closed in A (cid:2) ̟ (cid:3) G = A G (cid:2) ̟ (cid:3) , we conclude that( A G ) + = ( A + ) G . (cid:3) UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 27
Comparison of adic quotients and algebraic quotients.
For this section, we fix a com-plete, rank-1 valuation ring O K with fraction field K , and a choice of a pseudo-uniformizer ̟ .A rigid space over K always means here an adic space locally topologically finite type overSpa( K, O K ). When we need to use classical rigid-analytic spaces, we refer to them as Tate rigid-analytic spaces.In what follows, for any topologically finite type K -algebra A , we define Sp A := Spa( A, A ◦ ). Wenote that [Hub94, Lemma 4.4] implies that for any affinoid space Spa( A, A + ) that is topologicallyfinite type over Spa( K, O K ), we have A + = A ◦ . So this notation does not cause any confusion.We consider the analytification functor:( − ) an : (cid:26) locally finite type K -schemes (cid:27) → (cid:26) Rigid Spacesover K (cid:27) that is defined as a composition ( − ) an = r K ◦ ( − ) rig of the classical analytification functor ( − ) rig (as it is defined in [Bos14, § r K that sends a Tate rigid space to the associated adic space (see [Hub94, § X = Spec K [ T , . . . , T d ] /I iscanonically isomorphic to ∞ [ n =0 Sp (cid:18) K h ̟ n T , . . . , ̟ n T d i I · K h ̟ n T , . . . , ̟ n T d i (cid:19) (9)by the discussion after the proof of [Bos14, Lemma 5.4/1]. In particular, A n, an K is not affinoid as itis not quasi-compact. Lemma 4.5.1.
Let X be a rigid space over K with a K -action of a finite group G . Suppose thateach point x ∈ X admits an affinoid open neighborhood V x containing G.x . Then, for any classicalpoint x ∈ X/G , the natural map O X/G,x → Y x ∈ π − ( x ) O X,x G . is an isomorphism. Proof.
Theorem 4.3.4 gives that
X/G is a rigid space over K . Lemma 4.3.3 implies that we canassume that X = Sp A is an affinoid, so X/G ≃ Sp A G by Theorem 4.2.6 ([Hub94, Lemma 4.4]guarantees the equality of +-rings).Now, [Bos14, Corollary 4.1/5] implies that A G → O X/G,x is flat. Therefore, Lemma 2.1.12ensures that O X/G,x ≃ ( A ⊗ A G O X/G,x ) G . Finally, we note that the natural map A ⊗ A G O X/G,x → Y x ∈ π − ( x ) O X,x is an isomorphism by [Con06, Lemma A.1.3]. (cid:3)
Corollary 4.5.2.
Let X be a rigid space over K with a K -action of a finite group G . Supposethat each point x ∈ X admits an affinoid open neighborhood V x containing G.x . Then, for anyclassical point x ∈ X/G , the natural map b O X/G,x → Y x ∈ π − ( x ) b O X,x G . is an isomorphism. Proof.
By [Con06, Lemma A.1.3], each O X,x is O X/G,x -finite. Therefore, O X,x / m x O X,x is an artinian k ( x )-algebra. Thus, there is n x such that m n x x ⊂ m x O X,x . This implies that the m x -adic topologyon O X,x coincides with the m x O X,x -adic topology.Therefore, using that O X/G,x is noetherian [Bos14, Proposition 4.1/6] and O X,x is finite as an O X/G,x -module, we conclude that Y x ∈ π − ( x ) b O X,x ≃ Y x ∈ π − ( x ) O X,x ⊗ O X/G,x b O X/G,x . The claim now follows from Lemma 2.1.12 and Lemma 4.5.1. (cid:3)
Theorem 4.5.3.
Let X be a locally finite type K -scheme with a K -action of a finite group G .Suppose that any orbit G.x ⊂ X lies in an affine open subset. Then the analytification X an withthe induced K -action of G satisfies the assumption of Theorem 4.3.4, and the natural morphism: φ : X an /G → ( X/G ) an is an isomorphism. Proof. Step 1. The condition of Theorem 4.3.4 is satisfied for X an with the induced action of G :Firstly, Lemma 2.1.14 says that our assumption on X implies that there is a covering of X = ∪ i ∈ I U i by affine open G -stable subschemes. Then X an = ∪ i ∈ I U an i is an open covering of by G -stable adicsubspaces.Now we choose i such that x ∈ U an i . We note that (9) implies that each U an i can be written as aunion U an i = ∞ [ n =0 U ( n ) i of open affinoid subspaces. Since the orbit G.x is finite, it is contained in some U ( n ) i . Step 2. We reduce to a claim on completed local rings : We consider the commutative diagram: X an X an /G ( X/G ) an . π X an π an X φ Since π an X is a finite, surjective, G -equivariant morphism, we conclude that φ is finite, surjectivemorphism by Proposition 5.3.1. Therefore, [Con06, Lemma A.1.3] ensures that it suffices to showthat the natural map φ x : O ( X/G ) an ,φ ( x ) → O X an /G,x UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 29 is an isomorphism at each classical point of X an /G . We note that φ x is a local homomorphism ofnoetherian local rings by [Bos14, Proposition 4.1/6]. Thus, [Bou98, Chap III, § b φ x : b O ( X/G ) an ,φ ( x ) → b O X an /G,x is an isomorphism. Step 3. We show that b φ x is an isomorphism : We note Corollary 4.5.2 gives that b O X an /G,x ∼ = Y x ∈ π − X an ( x ) b O X an ,x G . Now we consider the natural morphism of locally ringed spaces i : ( X / G ) an → X / G . By [Con99,Lemma A.1.2(2)], i is a bijection between the sets of classical points of ( X/G ) an and closed pointsof X/G . Furthermore, the natural morphism b O X/G, i ( y ) → b O ( X/G ) an , y is an isomorphism for any closed point y ∈ ( X/G ) an .We denote z := i ( φ ( x )). Using finiteness of X → X/G and Lemma 2.1.12 we see that b O ( X/G ) an ,φ ( x ) ≃ b O X/G,z ≃ Y z ∈ π − X ( z ) b O X,z G , and φ x identified with the natural map Y z ∈ π − X ( z ) b O X,z G → Y x ∈ π − X an ( x ) b O X an ,x G . (10)Finally, we use [Con99, Lemma A.1.2(2)] once again to conclude that (10) is an isomorphism. (cid:3) Properties of the Geometric Quotients
We discuss some properties of schemes (resp. formal schemes, resp. adic spaces) that arepreserved by taking geometric quotients. For instance, one would like to know that
X/G is separated(resp. quasi-separated, resp. proper) if X is. This is not entirely obvious as X/G is explicitlyconstructed only in the affine case, and in general one needs to do some gluing to get
X/G . Thisgluing might, a priori, destroy certain global properties of X such as separatedness. In this sectionwe show that this does not happen for many geometric properties in all schematic, formal and adicsetups. We mostly stick to the properties we will need in our paper [Zav20]. One can repeat the proof of Corollary 4.5.2 using that Lemma 4.5.1 holds on the level of henselian local ringsin the scheme world.
Properties of the Schematic Quotients.
In this section, we discuss the schematic case.For the rest of the section, we fix a valuation ring k + . The proofs are written so they will adapt toother settings (formal schemes and adic spaces).Let f : X → Y be a G -invariant morphism of flat, locally finite type k + -schemes. We assumethat any orbit G.x ⊂ X lies inside some open affine subscheme V x ⊂ X . In particular, theconditions of Theorem 2.1.16 are satisfied, so it gives that X/G is a k + -scheme with a finitemorphism π : X → X/G . The universal property of the geometric quotient implies that f factorsthrough π and defines the commutative square: XX/G Y. π ff ′ Proposition 5.1.1.
Let f : X → Y , a finite group G , and f ′ : X/G → Y be as above. Then f ′ isquasi-compact (resp. quasi-separated, resp. separated, resp. proper, resp. finite) if f is so. Proof.
We note that all these properties are local on Y . Since the formation of X/G commuteswith open immersions, we can assume that Y is affine. Quasi-compactness : We suppose that f is quasi-compact. Using the fact that Y is affine, we seethat quasi-compactness of f (resp. f ′ ) is equivalent to quasi-compactness of the scheme X (resp. X/G ). Thus, X is quasi-compact. Since π is surjective by Theorem 2.1.16, we conclude that X/G is quasi-compact. Therefore, f ′ is quasi-compact as well. Quasi-separated : We suppose that the diagonal morphism ∆ X : X → X × Y X is quasi-compactand consider the following commutative square: X X × Y XX/G X/G × Y X/G ∆ X π π × Y π ∆ X/G
We know that π is finite, so it is quasi-compact. Therefore, the morphism π × Y π is quasi-compactas well, this implies that the morphism( π × Y π ) ◦ ∆ X = ∆ X/G ◦ π is quasi-compact. But we also know that π is surjective, so we see that quasi-compactness of∆ X/G ◦ π implies quasi-compactness of ∆ X/G . Thus f ′ is quasi-separated. Separatedness:
We consider the commutative square
X X × Y XX/G X/G × Y X/G ∆ X π π × Y π ∆ X/G
Since π is finite we conclude that π × Y π is finite as well, so it closed. Now we use surjectivity of π to get an equality: ∆ X/G ( X/G ) = ( π × Y π )(∆ X ( X ))with ( π × Y π )(∆ X ( X )) being closed as image of the closed subset ∆ X ( X ) (by separateness of X overaffine Y ). This shows that ∆ X/G ( X/G ) is a closed subset of
X/G × Y X/G , so
X/G is separated.
UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 31
Properness:
We already know that properness of f implies that f ′ is quasi-compact and sepa-rated. Also, Theorem 2.2.6 shows that f ′ is locally of finite type, so it is of finite type. The onlything that we are left to show is that it is universally closed. But this easily follows from universalclosedness of f and surjectivity of π . Finiteness:
A finite morphism is proper, so the case of proper morphisms implies that f ′ isproper. It is also quasi-finite as π is surjective and f = f ′ ◦ π has finite fibers. Now Zariski’s MainTheorem [EGA IV , Corollaire 18.12.4] implies that f ′ is finite. (cid:3) We now slightly generalize Proposition 5.1.1 to the case of a G -eqivariant morphism f . Namely,we consider a G -equivariant morphism f : X → Y of flat, locally finite type k + -schemes. We assumethat the actions of G on both X and Y satisfy the condition of Theorem 2.2.6. Then the universalproperty of the geometric quotient implies that f descends to a morphism f ′ : X/G → Y /G over k + . We show that various properties of f descend to f ′ : Proposition 5.1.2.
Let f : X → Y , a finite group G , and f ′ : X/G → Y /G be as above. Then f ′ is quasi-compact (resp. quasi-separated, resp. separated, resp. proper, resp. k -modification ,resp. finite) if f is so. Proof.
We start the proof by considering the commutative diagram
X YX/G Y /G fπ X π Y f ′ We denote by h : X → Y /G the composition f ′ ◦ π X = π Y ◦ f . Note that, for all properties P mentioned in the formulation of the proposition, f satisfies P implies that h satisfies P due tofiniteness of π Y . All but the k -modification property follow from Proposition 5.1.1 applied to h .Now suppose that f is a k -modification. We have already proven that f ′ is a proper map,so we only need to show that its restriction to k -fibers is an isomorphism. This follows fromthe fact that the formation of the geometric quotient commutes with flat base change such asSpec k → Spec k + . (cid:3) Lemma 5.1.3.
Let Y be a flat, locally finite type k + -scheme, and f : X → Y a G -torsor for afinite group G . The natural morphism f ′ : X/G → Y is an isomorphism. Proof.
Since a G -torsor is a finite ´etale morphism, we see that X is a flat, locally finite type k + -scheme. Moreover, we note that the conditions of Theorem 2.2.6 are satisfied as f is affine and theaction on Y is trivial. Thus, X/G is a flat, locally finite type k + -scheme. The universal propertyof the geometric quotient defines the map X/G → Y that we need to show to be an isomorphism. It suffices to check this ´etale locally on Y as theformation of X/G commutes with flat base change Theorem 2.1.16(4). Therefore, it suffices toshow that it is an isomorphism after the base change along X → Y . Now, X × Y X is a trivial G -torsor over X , so it suffices to show the claim for a trivial G -torsor. This is essentially obviousand follows either from the construction or from the univerisal property. (cid:3) A morphism f : X → Y of flat, locally finite type k + -schemes is called a k -modification , if it is proper and thebase change f k : X k → Y k is an isomorphism. Properties of the Formal Quotients.
Similarly to Section 5.1, we discuss that certainproperties descend through the geometric quotient in the formal setup. Most proofs are similar tothat in Section 5.1.For the rest of the section, we fix a complete, microbial valuation ring k + with a pseudo-uniformizer ̟ and field of fractions k .We consider a G -equivariant morphism f : X → Y of flat, locally finite type k + -schemes. Weassume that the actions of G on both X and Y satisfy the condition of Theorem 3.3.4. Then theuniversal property of the geometric quotient implies that f descends to a morphism f ′ : X /G → Y /G over k + : X YX /G Y /G. π X f π Y f ′ We show that various properties of f descend to f ′ : Proposition 5.2.1.
Let f : X → Y , a finite group G , and f ′ : X /G → Y /G be as above. Then f ′ is quasi-compact (resp. quasi-separated, resp. separated, resp. proper, resp. rig-isomorphism ,resp. finite) if f is so. Proof.
We note that in the case of a quasi-compact (resp. quasi-separated, resp. separated, resp.proper) f , the proof of Proposition 5.1.2 works verbatim. We only need to use Theorem 3.3.4 inplace of Theorem 2.2.6.The rig-isomorphism case is easy, akin to the k -modification case in of Proposition 5.1.2. Weonly need to use Theorem 4.4.1 in place of Theorem 2.1.16(4).Now suppose that f is finite. The proper case implies that f ′ is proper, and it is clearly quasi-finite. Therefore, the mod- ̟ fiber f ′ : ( X /G ) → ( Y /G ) is finite. Now [FK18, Proposition I.4.2.3]gives that f ′ is finite. (cid:3) Lemma 5.2.2.
Let Y be an admissible formal k + -scheme, and f : X → Y a G -torsor for a finitegroup G . The natural morphism f ′ : X /G → Y is an isomorphism. Proof.
The proof of Lemma 5.1.3 adapts to this situation. The only non-trivial fact that we usedis that one can check that a morphism is an isomorphism after a finite ´etale base change (and weuse Theorem 3.3.4(4) in place of Theorem 2.1.16(4)). This follows from descent for adic, faithfullyflat morphisms [FK18, Proposition I.6.1.5] (cid:3) Properties of the Adic Quotients.
Similarly to Section 5.1 and Section 5.2, we discussthat certain properties descend through the geometric quotient in the adic setup.For the rest of the section, we fix a locally strongly noetherian analytic space S .We consider a G -equivariant S -morphism f : X → Y of locally topologically finite type adic S -spaces. We assume that the actions of G on both X and Y satisfy the condition of Theorem 4.3.4. A morphism f : X → Y of admissible formal k + -schemes is called a rig-isomorphism if the adic generic fiber f k : X k → Y k is an isomorphism. The case of a finite flat morphism is much easier as the completed tensor product along a finite module coincideswith the usual tensor product due to Lemma 3.1.6(1).
UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 33
Then the universal property of the geometric quotient implies that f descends to a morphism f ′ : X/G → Y /G over S : X YX/G Y /G. π X f π Y f ′ We show that various properties of f descend to f ′ : Proposition 5.3.1.
Let f : X → Y , a finite group G , and f ′ : X/G → Y /G be as above. Then f ′ is quasi-compact (resp. quasi-separated, resp. separated, resp. proper, resp. finite) if f is so. Proof.
The proof is almost identical to that of Proposition 5.1.2. We use Theorem 4.3.4 in place ofTheorem 2.2.6 (that is used in the proof of Proposition 5.1.1 that Proposition 5.1.2 relies on). Weuse [Hub96, Proposition 1.5.5] in place of [EGA IV , Corollaire 18.12.4] to ensure that a quasi-finite,proper morphism is finite. (cid:3) Lemma 5.3.2.
Let Y be a locally topologically finite type adic S -space, and f : X → Y a G -torsorfor a finite group G . The natural morphism f ′ : X/G → Y is an isomorphism. Proof.
The proof of Lemma 5.1.3 adapts to this situation. The only non-trivial fact that we usedis that one can check that
X/G → Y is an isomorphism after a surjective, flat base change (andwe use Theorem 4.3.4(4) in place of Theorem 2.1.16(4)). This follows from Lemma B.4.6 as f ′ isfinite due to Proposition 5.3.1. (cid:3) AppendixAppendix A. Adhesive Rings and Boundedness of Torsion Modules
Let A be an ring with an ideal I . We define the notion of I -torsion part of an A -module anddiscuss some of its trivial properties. Then we define the notion of (universally) adhesive andtopologically (universally) adhesive rings. This Appendix does not prove any original results, butrather summarizes the main results from [FK18] in the form convenient for the reader.A.1. I -torsion Submodule.Definition A.1.1. Let M be an A -module, a ∈ A , and I ⊂ A an ideal.An element m ∈ M is a -torsion if a n m = 0 for some n ≥
1. The set of all a -torsion elements isdenoted by M a − tors .An element m ∈ M is I -torsion if m is a -torsion for any a ∈ I . The set of all I -torsion elementsis denoted by M I − tors .We say that M is I-torsion free if M I − tors = 0.An A -submodule N ⊂ M is saturated if M/N is I -torsion free. Remark A.1.2.
Suppose that
I, J ⊂ A are finitely generated ideals of such that I n ⊂ J and J m ⊂ I for some integers n and m . Then M I − tors = M J − tors for any A -module M . Lemma A.1.3.
Let A → B be a flat morphism, and I ⊂ A a finitely generated ideal, and M an A -module. Then M I − tors ⊗ A B ≃ ( M ⊗ A B ) IB − tors . Proof.
We start by choosing some generators I = ( a , . . . , a n ). Then M I − tors = \ M a i − tors , and ( M ⊗ A B ) IB − tors = \ ( M ⊗ A B ) a i − tors . Therefore, it suffices to show that M a − tors commutes with flat base change. Now we note that M a − tors = ∪ n M [ a n ] where M [ a n ] is a submodule of elements annihilated by a n . It is clear that( M ⊗ A B )[ a n ] = M [ a n ] ⊗ A B since B is A -flat. This implies that M a − tors = ( M ⊗ A B ) a − tors . (cid:3) Lemma A.1.3 implies that the notion of M I,tors can be globalized.
Definition A.1.4.
Let X be scheme, I a quasi-coherent ideal of finite type, and M a quasi-coherent O X -module. The O X -submodule of I -torsion elements M I − tors is defined as the sheafification of U M ( U ) I − tors . Remark A.1.5.
Lemma A.1.3 implies that M I ,tors is a quasi-coherent O X -module. If X = Spec A , I = e I , and M = f M . Then M I − tors ≃ ^ M I − tors . Definition A.1.6.
Let X be a scheme, and I a quasi-coherent ideal of finite type. We say that X is I -torsion free if O X,I − tors ≃ f : X → Y be a morphism of schemes, and I ⊂ O Y a quasi-coherent ideal of finite type. Wesay that X is I -torsion free if X is IO X -torsion free. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 35
A.2.
Universally Adhesive Schemes.Definition A.2.1.
A pair (
R, I ) of a ring and a finitely generated ideal is adhesive (or R is I -adically adhesive ) if Spec R \ V( I ) is noetherian and, for any finite R -module M , the I -torsionsubmodule M I − tors (see Definition A.1.1) is R -finite (see [FK18, Definition 0.8.5.4]).A pair ( R, I ) is universally adhesive if ( R [ X , . . . , X d ] , IR [ X , . . . , X d ]) is an adhesive pair for all d ≥ Remark A.2.2.
A valuation ring k + is universally adhesive if it is microbial (in the sense ofDefinition 3.1.1). More precisely, k + is universally ̟ -adically adhesive for any choice of a pseudo-uniformizer ̟ ∈ k + . Indeed, [FK18, Proposition 0.8.5.3] implies that it is sufficient to see that anyfinite k + [ X , . . . , X d ]-module M that is ̟ -torsion free (see Definition A.1.1) is finitely presented.This follows from Lemma 2.2.2(2) and the observation that M is torsion free if and only if it is ̟ -torsion free. Lemma A.2.3.
A valuation ring k + is I -adically adhesive for some finitely generated ideal I ifand only if k + is microbial. Proof. If k + is microbial, we take I = ( ̟ ) for any pseudo-uniformizer ̟ . Then k + is I -adicallyadhesive by Remark A.2.2.Now we suppose that k + is adhesive for some finitely generated ideal I . Then I = ( a ) isprincipal because k + is a valuation ring. Hence, k + (cid:2) a (cid:3) is a noetherian valuation ring by the I -adicadhesiveness of k + . Therefore, k + (cid:2) a (cid:3) is either a field or discrete valuation ring.We firstly consider the case k + (cid:2) a (cid:3) is a field. We then observe that rad( a ) is a height-1 primeideal of k + by [FK18, Proposition 0.6.7.2 and Proposition 0.6.7.3]. Therefore, k + is microbial by[Sem15, Proposition 9.1.3].Now we consider the case k + (cid:2) a (cid:3) a discrete valuation ring. Its maximal ideal m is clearly of height-1, so it defines a height-1 prime ideal p of k + . Hence, loc. cit. implies that k + is microbial. (cid:3) Here we summarize the main results about universally adhesive pairs:
Lemma A.2.4.
Let (
R, I ) be a universally adhesive pair, A a finite type R -algebra, and a finite A -module M . Then(1) Let N ⊂ M be a saturated A -submodule of M . Then N is a finite A -module.(2) If M is I -torsion free as an A -module. Then it is is a finitely presented A -module.(3) If A is I -torsion free as an R -module. Then it is a finitely presented R -algebra. Proof.
We choose some surjective morphism ϕ : R [ X , . . . , X d ] → A . Then the definition of univer-sal adhesiveness says that R [ X , . . . , X d ] is I -adically adhesive. This easily implies that so is A .Now the first two claims follow [FK18, Proposition 8.5.3]. To show the last claim, we note that thekernel ϕ is a saturated submodule of R [ X , . . . , X d ], so it is a finitely generated ideal by Part (1).Therefore, A is finitely presented as an R -algebra. (cid:3) Lemma A.2.5.
Let (
R, I ) be a universally adhesive pair (see Definition A.2.1), and A → B be afinite injective morphism of R -algebras. Suppose that B is of finite type over R , and that A ⊂ B is saturated (see Definition A.1.1). Then A is a finite type R -algebra. Proof.
The only non-formal part of the proof of Lemma 2.2.3 is to show that A is finite over A ′ .However, this follows from Lemma A.2.4. (cid:3) Corollary A.2.6.
Let (
R, I ) be a universally adhesive pair, and A an I -torsion free, finite type R -algebra with an R -action of a finite group G . The R -flat A G is a finite type R -algebra, and thenatural morphism A G → A is finitely presented. Proof.
The proof of Corollary 2.2.4 works verbatim. One only has to use Lemma A.2.5 in place ofLemma 2.2.3. (cid:3)
Definition A.2.7.
A pair ( X, I ) of a scheme and a quasi-coherent ideal of finite type is universallyadhesive if there is an open affine covering of X = ∪ i ∈ I Spec U i such that ( O ( U i ) , I ( U i )) is universallyadhesive for all i ∈ I . Remark A.2.8.
The notion of universal adhesiveness is independent of a choice of affine opencovering. This is explained in [FK18, Proposition 8.5.6 and Proposition 8.6.7]. It essentially followsfrom Lemma A.1.3 and that noetherianness is local in fppf topology.
Theorem A.2.9.
Let ( S, I ) be a universally adhesive pair (in the sense of Definition A.2.7), and X be an I -torsion free, locally finite type S -scheme with an S -action of a finite group G . Supposethat each point x ∈ X admits an affine neighborhood V x containing G.x . Then the scheme
X/G as in Theorem 2.1.16 is I -torsion free and locally finite type over S , and the integral surjection π : X → X/G is finite and finitely presented.
Proof.
The proof of Theorem 2.2.6 just goes through if one uses Corollary A.2.6 instead of Corol-lary 2.2.4. (cid:3)
A.3.
Universally Adhesive Formal Schemes.Definition A.3.1.
A pair (
R, I ) of a ring and a finitely generated ideal is topologically universallyadhesive (or R is I -adically topologically universally adhesive ) if ( R, I ) is universally adhesive, andthe pair ( b R h X , . . . , X d i , I b R h X , . . . , X d i ) is adhesive (in the sense of Definition A.2.1) for any d ≥ R endowed with the adic topology defined by a finitely generatedideal of definition I ⊂ R is topologically universally adhesive if R is I -adically complete, and thepair ( R, I ) is topologically universally adhesive.
Remark A.3.2.
We note that the definition of topologically universally adhesive topological ringsis independent of a choice of a finitely generated ideal of definition. For any two ideal of definition I and J , I n ⊂ J and J m ⊂ I for some integers n and m . Therefore, M I − tors = M J − tors byRemark A.1.2. Remark A.3.3.
We note that a microbial valuation ring k + is topologically universally adhesive.More precisely, k + is topologically universally ̟ -adically adhesive for any choice of a pseudo-uniformizer ̟ ∈ k + . This is proven in [FK18, Theorem 0.9.2.1]. Alternatively, one can easily showthe claim from Lemma 3.1.6 and the classical fact that k is strongly noetherian, i.e. k h X , . . . , X d i is noetherian for any d ≥ Lemma A.3.4.
Let R be a complete, topologically universally I -adically adhesive ring, A be atopologically finite type R -algebra, and M a finite A -module. Then(1) M is I -adically complete. In particular, A is I -adically complete.(2) Let N ⊂ M be an A -submodule of M . Then the I -adic topology on M restricts to the I -adic tpopology on N .(3) Let N ⊂ M be a saturated A -submodule of M . Then N is a finite A -module. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 37 (4) If M is R -flat, it is finitely presented over A .(5) If A is R -flat, it is topologically finitely presented.(6) For any element f ∈ A , the completed localization A { f } = lim n A f /I n A f is A -flat. Proof.
The first claim is proven in [FK18, Proposition 0.8.5.16 and Proposition 7.4.11]. The secondclaim is [FK18, Proposition 0.8.5.16]. The proofs of Parts (3)-(5) are similar to the proof ofanalogous statements in Lemma A.2.4. The last Part is proven in [FK18, Proposition I.2.1.2]. (cid:3)
Definition A.3.5.
An algebra A over a complete, topologically universally I -adically adhesive ring R is admissible if A is topologically finite type over R and I -torsion free. Lemma A.3.6.
Let R be an I -adically complete, I -adically topologically universally adhessivering (see Definition A.3.1), and A → B be a finite injective morphism of I -adically complete R -algebras. Suppose that B is topologically finite type over R , and A ⊂ B is saturated in B (SeeDefinition A.1.1). Then A is a topologically finite type R -algebra. Proof.
The main non-formal part of the proof of Lemma 3.2.3 is to show that A is finite over A ′ .This follows from A.3.4. (cid:3) Corollary A.3.7.
Let R be an I -adically complete, I -adically topologically universally adhessivering, and A an admissible R -algebra (in the sense of Definition A.3.5) with an R -action of a finitegroup G . Then A G is an admissible R -algebra, the induced topology on A G coincides with the I -adic topology, and A is finitely presented A G -module. Proof.
The proof of Corollary 3.2.4 works verbatim. One only needs to use Lemma A.3.6 in placeof Lemma 3.2.3 and Lemma A.3.4 in place of Lemma 3.1.6. (cid:3)
Proposition A.3.8.
Let R be an I -adically complete, I -adically topologically universally adhessivering, and X = Spf A an affine admissible formal R -scheme with an R -action of a finite group G .Then the natural map φ : X /G → Y = Spf A G is an R -isomorphism of topologically locally ringedspaces. In particular, X /G is an admissible formal R -scheme. Proof.
The proof of Proposition 3.2.6 goes through in this more general set-up. The only two differ-ences are that one needs to deduce that A G is admissible over R (with the induced topology equalsto the I -adic) from Corollary A.3.7 instead of Corollary 3.2.4 and one needs to use Lemma A.3.4(6)instead of Lemma 3.1.6(6) to ensure that ( A G ) { f } is an A G -flat module. (cid:3) Definition A.3.9.
A formal scheme X is locally universally adhesive if there exists an affine opencovering X = ∪ i ∈ I U i such that each U i is isomorphic to Spf A with A a topologically universallyadhesive ring. If X is, moreover, quasi-compact, we say that X is universally adhesive. Remark A.3.10.
Definition A.3.9 is independent of a choice of open covering. More precisely, anaffine formal scheme X = Spf A is universally adhesive if and only if A is topologically universallyadhesive. This is shown in [FK18, Propostition 2.1.9]. Remark A.3.11.
Lemma A.3.4(6) can be strengthen to the statement that an adic morphism ofaffine universally adhesive formal schemes Spf B → Spf A is flat if and only if A → B is flat. Thisis proven from [FK18, Proposition I.4.8.1]. We follow [FK18] and say that an adic morphism of formal schemes f : X → Y is flat if and only if O Y ,f ( x ) → O X ,x is flat for all x ∈ X . Definition A.3.12.
Let S be a universally adhesive formal scheme. An adic S -scheme X is called admissible if it is locally of topologically finite type, and there is an affine open covering X = ∪ i ∈ I U i such that each U i is isomorphic to Spf A with A an I -torsion free ring for a(ny) finitely generatedideal of definition I ⊂ A .We show that this definition is independent of a choice of a covering. Lemma A.3.13.
Let X = Spf A be an affine, locally of topologically finite type formal S -scheme.Then X is admissible if and only if A is I -torsion free for a(ny) finitely generated ideal of definition I . Proof.
First of all, we note that Remark A.1.2 implies that Definition A.3.12 is independent ofa choice of a finitely generated ideal of definition I . Thus, using that X is quasi-compact, wecan assume that X = ∪ ni =1 U i = Spf A i with A i an I -torsion free A -algebra. Then the morphism A → Q ni =1 A i is faithfully flat by Remark A.3.11 and the fact that all maximal ideals are open inan I -adically complete ring (see [FK18, Lemma 0.7.2.13]). Now Lemma A.1.3 implies that n Y i =1 A i ! I − tors ≃ A I − tors ⊗ A n Y i =1 A i ! . Our assumption implies that ( Q ni =1 A i ) I − tors ≃
0. Therefore, A I − tors ≃ A → Q ni =1 A i isfaithfully flat. (cid:3) Lemma A.3.14.
Let S be a universally adhesive formal scheme, and let X be an S -finite, admis-sible formal S -scheme. Suppose than S ′ → S is an adic morphism of universally adhesive formalscheme. Then X ′ := X × S S ′ is an admissible formal S ′ -scheme. Proof.
Lemma A.3.13 ensures that the question is Zariski local on X ′ . Thus, we may and do assumethat S , S ′ (and, therefore, X ) are affine. Suppose S = Spf A , S ′ = Spf A ′ , and X = Spf B forsome finite A -module B (see [FK18, Proposition I.4.2.1]). Choose an ideal of definition I ⊂ A , ourassumptions imply that IA ′ is an ideal of definition in A ′ . We know that X ′ is given by Spf A ′ b ⊗ A B .We note that A ′ ⊗ A B is finite over A ′ , so it is already IA ′ -adically complete by Lemma A.3.4(1).Therefore, we conclude that X ′ ≃ Spf A ′ ⊗ A B . Now the claim follows from Lemma A.3.14 andRemark A.3.11. (cid:3) Theorem A.3.15.
Let S be a universally adhesive formal scheme (see Definition A.3.9), and X an admissible formal S -scheme (see Definition A.3.12). Suppose that X has an S -action of a finitegroup G such that each point x ∈ X admits an affine neighborhood V x containing G.x . Then X /G is an admissible formal S -scheme. Moreover, it satisfies the following properties:(1) π : X → X /G is universal in the category of G -invariant morphisms to topologically locallyringed S -spaces.(2) π : X → X /G is a finite, surjective, topologically finitely presented morphism (in particular,it is closed).(3) Fibers of π are exactly the G -orbits.(4) The formation of the geometric quotient commutes with flat base change, i.e. for anyuniversally adhesive formal scheme Z and a flat adic morphism Z → X /G , the geometricquotient ( X × X /G Z ) /G is a formal schemes, and the natural morphism ( X × X /G Z ) /G → Z is an isomorphism. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 39
Proof.
The proofs of parts (1), (2), and (3) are similar to that of Theorem 3.3.4. The main differenceis that one needs to use Proposition A.3.8 in place of Proposition 3.2.6, Lemma A.3.4 in place ofLemma 3.1.6, and [FK18, Proposition I.2.2.3] in place of [Bos14, Proposition 7.3/10].We explain part (4) in a bit more detail. We firstly reduce to the case S = Spf R , X = Spf A with A a finite R -module, and S ′ = Spf R ′ . Then R → R ′ is flat by Remark A.3.11. ThenLemma A.3.14 implies that X ′ is S ′ -admissible, and then one can repeat the proof of Theorem 3.3.4using Lemma A.3.4(1) in place of Lemma 3.1.6(1). (cid:3) Theorem A.3.16.
Let R be an topologically universally I -adically adhesive ring, and X an I -torsion free, locally finite type R -scheme with a R -action of a finite group G . Suppose that anyorbit G.x ⊂ X lies in an affine open subset V x . The same holds for its I -adic completion b X withthe induced b R + -action of G , and the natural morphism b X/G → [ X/G is an isomorphism.
Proof.
The proof of Theorem 3.4.1 goes through in this wider generality. The only new non-trivialinput is flatness of A G → c A G . More generally, this flatness holds for any finite type R -algebra B . Namely, any such algebra is I -adically adhesive, so it satisfies the so called BT property (seeDefinition [FK18, Section 0.8.2(a)]) by [FK18, Proposition 0.8.5.16]. Therefore, [FK18, 8.2.18(i)]implies that B → b B is flat. (cid:3) Appendix B. Foundations of Adic Spaces
The theory of adic spaces still seems to lack a “universal reference” for proofs of all basic questionsone might want to use. For example, all of [Hub93a], [Hub94], [Hub96] and [KL16] do not reallydiscuss notions of flat and separated morphisms in detail. The two main goals of this Appendix areto provide the reader with the main definitions we use in the paper, and to give proofs of claimsthat we need in the paper and that seem difficult to find in the standard literature on the subject.We stick to the case of analytic adic spaces since this is the only case that we need in thispaper.B.1. Basic Definitions.
We start this section by introducing two categories that will play the roleof the categories of (locally) ringed spaces in the case of adic spaces. Namely, these two categoriesare ambient categories for the category of analytic adic spaces, but the embedding of adic spacesin one of them is not fully faithful.
Definition B.1.1.
We recall the definition of the category of valuative topologically locally ringedspaces V from [Sem15, Definition 13.1.1] . The objects of this category are triples ( X, O X , { v x } x ∈ X )such that(1) X is a topological space,(2) O X is a sheaf of topological rings such that the stalk O X,x is a local ring for all x ∈ X ,(3) v x is a valuation on the residue field k ( x ) of O X,x .Morphisms f : X → Y of objects in V are defined as maps ( f, f ) of topologically locally ringedspaces such that the induced maps of residue fields k ( f ( x )) → k ( x ) are compatible with valuations(equivalently, induces a local inclusion between the valuation rings). We recall that an adic space X is required to be sheafy, i.e. the structure presheaf O X must be a sheaf It is slightly different from [Hub94, page 521].
Remark B.1.2.
The category V comes with the forgetful functor F : V → TLRS to the categoryof topologically locally ringed spaces. It is clear to see that this functor is conservative.
Definition B.1.3.
We define the category AS of analytic adic spaces as a full subcategory of V , whose objects are triples ( X, O X , { v x } x ∈ X ) locally isomorphic to Spa( A, A + ) for a completeTate-Huber pair ( A, A + ). We remind the reader that this requires the pair ( A, A + ) is “sheafy”. Remark B.1.4.
Given any analytic adic space ( X, O X , { v x } x ∈ X ), Huber defined a sheaf O + X asfollows O + X ( U ) = { f ∈ O X ( U ) | v x ( f ) ≤ x ∈ U } . We note that [Hub94, Proposition 1.6] implies that O + X ( X ) = A + for any sheafy complete Tate-Huber pair ( A, A + ) and X = Spa( A, A + ). Remark B.1.5.
Note that [Hub94, Proposition 2.1(ii)] guarantees that we have a natural identi-fication Hom AS ( X, Spa(
A, A + )) = Hom cont (( A, A + ) , ( O X ( X ) , O + X ( X ))) . for any analytic adic space X . Definition B.1.6. A Tate affinoid adic space is an object of the category AS that is isomorphicto Spa( A, A + ) for a complete Tate-Huber pair ( A, A + ) Remark B.1.7.
In general, there are analytic affinoid adic spaces that are not isomorphic toSpa(
A, A + ) for any complete Tate-Huber pair ( A, A + ). The analytic condition implies the existenceof a pseudo-uniformizer only locally on Spa( A, A + ), but it does not necessarily exist globally. See[Ked17, Example 1.5.7] for an explicit example of an analytic affinoid adic space that is not a Tateaffinoid.B.2. Finite and Topologically Finite Type Morphisms of Adic Spaces.Definition B.2.1.
We say that a morphism of complete Tate-Huber pairs (
A, A + ) → ( B, B + ) is topologically of finite type , if there is a surjective quotient map f : A h T , . . . , T n i → B such that B + is integral over A + h T , . . . , T n i . Remark B.2.2.
This definition coincides with the definition of topologically finite type morphismof Huber pairs from [Hub94, page 533]. This is stated in [Hub94, Lemma 3.3 (iii)] and it is provenin [Sem15, Proposition 15.3.3].
Remark B.2.3.
It turns out that any continuous surjective morphism f : C → B of complete Taterings is a quotient mapping. Moreover, it is actually an open map; this is the content of the BanachOpen Mapping Theorem [Hub94, Lemma 2.4 (i)].There are crucial properties of topologically finite type morphisms that makes it behave similarlyto the notion of finite type morphisms: Lemma B.2.4 ([Hub94]) . Let f : ( A, A + ) → ( B, B + ) and g : ( B, B + ) → ( C, C + ) be continuoushomomorphisms of complete Tate-Huber pairs. If f and g are topologically finite type morphismsthen so is g ◦ f , and if g ◦ f is topologically finite type then so is g . Proof.
This is proven in [Hub94, Lemma 3.3 (iv)]. (cid:3) This is the set of all continuous ring homomorphisms f : A → O X ( X ) such that f ( A + ) ⊂ O + X ( X ). We do notclaim that ( O X ( X ) , O + X ( X )) is a (Tate-)Huber pair. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 41
Definition B.2.5.
A morphism of analytic adic spaces f : X → Y is called locally of topologicallyfinite type , if there is an open covering of Y by Tate affinoids { V i } i ∈ I and an open covering of X by Tate affinoids { U i } i ∈ I such that f ( U i ) ⊂ V i , and ( O Y ( V i ) , O + Y ( V i )) → ( O X ( U i ) , O + X ( U i ))is topologically of finite type (in the sense of Definition B.2.1). If a morphism f is locally oftopologically finite type and quasi-compact, it is called topologically finite type .The relation of Definition B.2.5 to Definition B.2.1 for affinoid X and Y is addressed in Theo-rem B.2.18 under some noetherian condition. Definition B.2.6.
We say that a morphism of complete Tate-Huber pairs (
A, A + ) → ( B, B + ) is finite , if the ring homomorphism A → B is finite, and a ring homomorphism A + → B + is integral. Remark B.2.7.
Our definition coincides with the definition in Huber’s book [Hub96, (1.4.2)] dueto the following (easy) Lemma.
Lemma B.2.8.
A finite morphism of complete Tate-Huber pairs f : ( A, A + ) → ( B, B + ) is oftopologically finite type. Proof.
We choose a set ( y , . . . , y m ) of A -module generators for B . After multiplying by somepower of a pseudo-uniformizer ̟ we can assume that y i ∈ B + for all i . Then we use the universalproperty [Hub94, Lemma 3.5 (i)] to define a continuous surjective morphism g : A h T , . . . , T m i → B as a unique continuous A -linear homomorphism such that f ( T i ) = y i . It is easily seen to besurjective, and it is open by Remark B.2.3. Moreover, B is integral over A + h T , . . . , T m i since it iseven integral over A + by the definition of finiteness. (cid:3) Lemma B.2.9.
Let f : ( A, A + ) → ( B, B + ) be a topologically finite type morphism of completeTate-Huber pairs such that B + is integral over A + . Then there exist rings of definition A ⊂ A and B ⊂ B such that f ( A ) ⊂ B and B is finite over A . In particular, ( A, A + ) → ( B, B + ) isfinite. Proof.
We use Remark B.2.3 to find an open, surjective morphism h : A h T , . . . , T n i ։ B. Clearly B + is integral over A + h T , . . . , T n i . The topological generators b i := h ( T i ) ∈ B + are integralover A + .Pick monic polynomials F i ∈ A + [ T ] such that F i ( b i ) = 0 for all i . We look at the coefficients { a i,j } ∈ A + ⊂ A ◦ of the polynomials F i . There are only finitely many of them, so we claim thatwe can find a pair of definition ( A , ̟ ) ⊂ A + such that A contains every a i,j . Indeed, we pickany ring of definition A ′ in A + and consider the subring generated by A ′ and every a i,j . It is easyto see that the resulted ring is open and bounded in A , so it is a ring of definition by [Hub93a,Proposition 1.1].Now we define a ring of definition ( B , ̟ ) as the image h ( A h T , . . . , T n i ). It is open because h is open, and it is bounded because any morphism of Tate rings preserves boundedness.We claim that a natural morphism A → B is finite. It suffices to prove that it is finite mod ̟ by successive approximation and completeness. However, it is clearly finite type mod ̟ since itcoincides with the composition: A /̟A → ( A /̟A ) [ T , . . . , T n ] ։ B /̟B , and it is integral since B /̟B is algebraically generated over A /̟A by residue classes b , . . . , b n that are integral over A /̟A by construction. Thus this map is integral and finite type, hencefinite.Finally, ( A, A + ) → ( B, B + ) is finite since A → B is equal to the finite map A (cid:20) ̟ (cid:21) → B (cid:20) ̟ (cid:21) . (cid:3) Remark B.2.10.
The proof of Lemma B.2.9 actually shows more. We can choose B to containany finite set of elements x , . . . , x m ∈ B + . Indeed, the proof just goes through if one replaces h : A h T , . . . , T n i → B at the beginning of the proof with the continuous A -algebra morphism h ′ : A h T , . . . , T n ih X , . . . , X m i → B satisfying h ′ ( T i ) = b i and h ′ ( X j ) = x j . Existence of such morphism follows from the universalproperty of restricted power series (see [Hub94, Lemma 3.5(i)]) Lemma B.2.11.
Let f : ( A, A + ) → ( B, B + ) be a finite morphism of complete Tate-Huber pairs.If f induces an isomorphism A ≃ B then f is an isomorphism of Tate-Huber pairs. Proof.
We note that B + is integral over A + by the definition of a finite morphism, and f is open byRemark B.2.3. However, A + is integrally closed in A = B . Thus the injective morphism A + → B + is an isomoprhism. (cid:3) Definition B.2.12.
A morphism of analytic adic spaces f : X → Y is called finite , if there is acovering of Y by Tate affinoids { V i } i ∈ I such that each f − ( V i ) is an open Tate affinoid subset of Y ,and a natural morphism ( O Y ( V i ) , O + Y ( V i )) → ( O X ( U i ) , O + X ( U i )) is finite (in the sense of DefinitionB.2.6) for all i .The relation between Definition B.2.12 and Definition B.2.6 in the case of affinoid X and Y isaddressed in Theorem B.2.18 under some noetherian constraints. Definition B.2.13.
A Tate-Huber pair (
A, A + ) is called strongly noetherian if A h T , . . . , T n i isnoetherian for all n . Lemma B.2.14.
Let (
A, A + ) be a strongly noetherian complete Tate-Huber pair. A topologicallyfinite type complete ( A, A + )-Tate-Huber pair ( B, B + ) is strongly noetherian as well. Proof.
This is proven in [Hub94, Corrolary 3.4] (cid:3)
Definition B.2.15.
An analytic adic space S is called locally strongly noetherian , if every point x ∈ S has an affinoid open neighborhood isomorphic to Spa( A, A + ) for some strongly noetheriancomplete Tate-Huber pair ( A, A + ). Remark B.2.16.
It is not known (to the author) if a strongly noetherian Tate affinoid adic spaceis always isomorphic to Spa(
A, A + ) for a strongly noetherian complete Tate-Huber pair ( A, A + ). Definition B.2.17.
A Tate affinoid Spa(
A, A + ) is called strongly noetherian if is isomorphic toSpa( A, A + ) for a strongly noetherian Tate-Huber pair ( A, A + ).Remark B.2.16 suggests that it might be difficult to control strong noetherianess of open affinoids(since we cannot check it locally). However, it turns out that it is possible to do, as we will see inCorollary B.2.21. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 43
Theorem B.2.18.
Let f : Spa( B, B + ) → Spa(
A, A + ) be a finite (resp. topologically finite type)morphism of strongly noetherian Tate affinoids. The corresponding map f : ( A, A + ) → ( B, B + )is finite (resp. topologically finite type). Proof.
This is proven in [Hub93b, p. 3.3.23] in the case of topologically finite type morphsisms, andit is proven in [Hub93b, p. 3.6.20] in the case of finite morphisms. (cid:3)
Theorem B.2.19.
Let (
A, A + ) be a strongly noetherian Tate-Huber pair, and f : X → Spa(
A, A + )a finite morphism. Then X is affinoid and the morphism ( A, A + ) → ( O X ( X ) , O + X ( X )) is finite. Proof.
This is proven in [Hub93b, p. 3.6.20]. (cid:3)
Remark B.2.20.
We do not know Theorem B.2.18 or Theorem B.2.19 hold without the extrastrong noetherianness assumption.
Corollary B.2.21.
Let (
A, A + ) be a strongly noetherian Tate-Huber pair, and let Spa( B, B + ) ⊂ Spa(
A, A + ) be an open affinoid subspace. Then ( B, B + ) is a strongly noetherian Tate-Huber pair. Proof.
An open immersion Spa(
B, B + ) ⊂ Spa(
A, A + ) is a morphism topologically of finite type.Thus we apply Theorem B.2.18 to see that ( A, A + ) → ( B, B + ) is topologically of finite type. SoLemma B.2.14 implies the results. (cid:3) B.3.
Completed Tensor Products.
The main goal of this section is to prove that under certainassumptions, completed tensor products of Tate rings coincide with usual tensor products. Thisshould be well-known to the experts, but it seems difficult to extract the proof from the existingliterature.For the rest of the section, we fix a complete Tate-Huber pair (
A, A + ) with a choice of a pair ofdefinition ( A , ̟ ). We recall the notion of “the natural A -module topology” for a finite A -module M : Definition B.3.1.
A topological A -module structure on M is natural if any A -linear map M → N to a topological A -module N is continuous.It is clear that the natural A -module topology is unique, if it exists. Using the universal propertyof direct products, it is easy to see that the natural A -module topology provides (if exists) a structureof a topological A -module on M . It turns out that the natural topology actually always exists. Lemma B.3.2.
Let M be a finite A -module. There is a topology on M that satisfies the definitionof the natural A -module topology. Proof.
First of all, we claim that the product topology on a finite free module A n is the natural A -module topology on it. Indeed, it suffices to proof the claim in the case n = 1 by the universalproperty of direct products. But any A -linear map A → N to a topological A -module is clearlycontinuous.Now we deal the case of an arbitrary finitely generated M . We choose a surjective morphism f : A n → M and provide M with the quotient topology. This is clearly a topological A -modulestructure. We want to show that any A -linear morphisms g : M → N to a topological A -module N is continuous. We consider the diagram: A n M N f hg
Then for any open U ⊂ N we see that f − ( g − ( U )) = h − ( U ) is open since h is continuous by theargument above. The definition of quotient topology implies that g − ( U ) is open as well. Thus g is indeed continuous. (cid:3) Remark B.3.3.
We warn the reader that a natural topology on a finite A -module may not becomplete as A may have non-closed ideals. Lemma B.3.4.
Let M be a finite complete A -module. The topology on M is the natural A -moduletopology. If ( B, B + ) is a finite complete ( A, A + )-Tate-Huber pair, then there is a ring of definition B and a surjective A -linear morphism p : A n → B with p ( A n ) = B . Proof.
In the case of a finite complete module M , any surjection A n → M must be open by [Hub94,Lemma 2.4(i)]. So M carries the natural A -module topology by the construction of that in thepoof of Lemma B.3.2.As for the second claim, we use Lemma B.2.9 to find rings of definition A , B such that B is finite over A . Choose some generators b , . . . , b n for B over A and consider the morphism p : A n → B that sends ( a , . . . , a n ) to a b + · · · + a n b n . Then clearly p ( A n ) = B . (cid:3) Finally, we recall that given two morphisms f : A → B and g : A → C of Tate rings there is acanonical way to topologize the tensor product B ⊗ A C . Namely, we pick some rings of definitions B ⊂ B and C ⊂ C such that f ( A ) ⊂ B and g ( A ) ⊂ C . Then we topologize A ⊗ B C byrequiring the image ( B ⊗ A C ) := Im( B ⊗ A C → B ⊗ A C ) with its ̟ -adic topology to be a ringof definition in B ⊗ A C . Huber shows in [Hub93b, p. 2.4.18] that this Tate ring satisfies the expecteduniversal property in the category of Tate rings. In particular, this shows that this constructiondoes not depend on the choice of rings of definitions A , B , C . But we warn the reader that B ⊗ A C need not be (separated and) complete even if A, B and C are; its completion is denotedby B b ⊗ A C . Lemma B.3.5.
Let f : ( A, A + ) → ( B, B + ) be a finite morphism of complete Tate-Huber pairs,and let g : A → C be any morphisms of Tate rings. Then the topologized tensor product B ⊗ A C has the natural C -module topology.We note that this is not automatic from Lemma B.3.4 since B ⊗ A C is not necessarily complete. Proof.
We use Lemma B.3.4 to find a ring of definition B ⊂ B and a surjection p : A n → B suchthat p ( A n ) = B . Then after tensoring it against C we get a surjective morphism C n → B ⊗ A C ,and tensoring the surjection A n → B against C we get a surjection C n → B ⊗ A C . Combiningthese, we get a commutative diagram: C n C n B ⊗ A C ( B ⊗ A C ) B ⊗ A C p C By definition, ( B ⊗ A C ) with its ̟ -adic topology is open in B ⊗ A C , so p C | C n : C n → B ⊗ A C is open onto an open image. Hence, p C is also open, so B ⊗ A C has the quotient topology via p C as desired. (cid:3) Lemma B.3.6.
Let f : ( A, A + ) → ( B, B + ) be a finite morphism of complete Tate-Huber pairswith noetherian A . Suppose that A → C is a continuous morphism of noetherian, complete Taterings. Then the natural morphism B ⊗ A C → B b ⊗ A C is a topological isomorphism. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 45
Proof.
Lemma B.3.5 implies that B ⊗ A C carries the natural C -module topology. Then we use[Hub94, Lemma 2.4(ii)] to conclude that B ⊗ A C is already complete, so the completion map B ⊗ A C → B b ⊗ A C is a topological isomorphism. (cid:3) Corollary B.3.7.
Let f : ( A, A + ) → ( B, B + ) be a finite morphism of complete Tate-Huber pairswith a strongly noetherian Tate ring A . Then the natural morphism B ⊗ A A (cid:28) f g , . . . , f n g (cid:29) → B (cid:28) f g , . . . , f n g (cid:29) is a topological isomorphism for any choice of elements f , . . . , f n , g ∈ A generating the unit idealin A . Proof.
First of all, we note that A D f g , . . . , f n g E is a complete Tate ring. Moreover, it is noetherianby [Hub94, (II.1), (iii) on page 530] so we can apply Lemma B.3.6 with C = A D f g , . . . , f n g E . Thusthe question is reduced to show that the natural morphism B b ⊗ A A (cid:28) f g , . . . , f n g (cid:29) → B (cid:28) f g , . . . , f n g (cid:29) is a topological isomorphism. But this easily follows from the universal properties of topologizedtensor products [Hub93b, p. 2.4.18], completions [Sem15, Proposition 7.2.2] and completed rationallocalizations [Hub94, Corollary 3.4]. (cid:3) B.4.
Flat Morphisms of Adic Spaces.
We discuss the notion of a flat morphism of adic spaces.This notion is not discussed much in the existing literature, so we provide the reader with somefacts that we are using in the paper.
Definition B.4.1.
A morphism of analytic adic spaces f : X → Y is called flat , if the naturalmorphism O Y,f ( x ) → O X,x is flat for any point x ∈ X .Similarly to the case of formal schemes, we will soon describe flatness of strongly noetherianTate affinoids in more concrete terms. Lemma B.4.2.
Let X = Spa( A, A + ) be a strongly noetherian Tate affinoid adic space, and let x ∈ X be a point corresponding to a valuation v with support p . Then the natural morphism r x : A p → O X,x is faithfully flat.
Proof.
We note that rational subdomains form a basis of topology on affinoid space, so O X,x is equalto a filtered colimit of O X ( U ) over all rational subdomains in X containing x . We use [Hub94,(II.1), (iv) on page 530] to note that A → O X ( U ) is flat for each such U . Since flatness is preservedby filtered colimits, we conclude that A → O X,x is flat. Note that this implies that A p → O X,x is flatas well. Indeed, this easily follows from the fact that for any A p -module M we have isomorphisms M ⊗ A p O X,x ∼ = ( M ⊗ A A p ) ⊗ A p O X,x ∼ = M ⊗ A O X,x . The discussion above shows that r x : A p → O X,x is flat, but we also need to show that it isfaithfully flat. In order to prove this claim it suffices to show that A p → O X,x is a local ringhomomorphism. Now we recall that the maximal ideal m x ⊂ O X,x is given as m x = { f ∈ O X,x | v ( f ) = 0 } We need to show r x ( p A p ) ⊂ m x . We pick any element h ∈ p A p . It can be written as f /s for f ∈ p and s ∈ A \ p , and we need to check that v (cid:16) fs (cid:17) = 0. The very definition of p as the support of v implies that v ( f ) = 0 and v ( s ) = 0. Thus v (cid:18) fs (cid:19) = v ( f ) v ( s ) = 0 . (cid:3) Lemma B.4.3.
Let f : Spa( B, B + ) → Spa(
A, A + ) be a flat morphism of strongly noetherian Tateaffinoid adic spaces. The natural morphism A → B is flat as well. Proof.
We start the proof by noting that [Hub94, Lemma 1.4] implies that for any maximal ideal m ⊂ B there is a valuation v ∈ Spa(
B, B + ) such that supp( v ) = m . It easy to see thatsupp ( w ) = ( f ) − ( m ) =: p where w = f ( x ) ∈ Spa(
A, A + ). We use Lemma B.4.2 to conclude that we have a commutativesquare B m O X,v A p O Y,wr m f p r p f w with r m and r p being faithfully flat. It is easy to see now that flatness of f w implies flatness of f p .Finally we note that m was an arbitrary maximal ideal in B , so A → B is flat. (cid:3) Remark B.4.4.
We warn the reader that it is unknown whether Lemma B.4.3 remains true if onedrops the strongly noetherian hypothesis. Even the case of rational embeddings is open. However,there are some positive results in this direction in [KL16, § Remark B.4.5.
We also do not know if flatness of A → B implies flatness of Spa( B, B + ) → Spa(
A, A + ) even in the strongly noetherian case. One can show by arguing over classical points that this holds in the case of A and B topologically finite type K -algebras over a complete, rank-1valued field K . Lemma B.4.6.
Let f : X = Spa( B, B + ) → Y = Spa( A, A + ) be a finite morphism of stronglynoetherian Tate affinoids, and g : Z = Spa( C, C + ) → Spa(
A, A + ) be a surjective flat morphism ofstrongly noetherian Tate affinoids. Then f is an isomorphism if and only if f ′ : X × Y Z → Z is. Proof.
We note that f ′ is finite by [Hub96, Lemma 1.4.5(i)], so Lemma B.2.11 ensures that itsuffices to show that A → B is a (topological) isomorphism if and only if C → C b ⊗ A B is. We canignore topologies by Remark B.2.3.Now we note that Lemma B.3.6 gives that C ⊗ A B ≃ C b ⊗ A B . Thus, it suffices to show that A → B is an isomorphism if and only if C → C ⊗ A B is. This follows from the usual faithfully flatdescent as A → C is flat by Lemma B.4.3, and therefore faithfully flat by [Hub94, Lemma 1.4]. (cid:3) B.5.
Coherent Sheaves.
We review the basic theory of coherent sheaves on locally strongly noe-therian adic spaces.We firstly recall the construction of an O X -module f M on a strongly noetherian Tate affinoid X = Spa( A, A + ) associated to a finite A -module M . For each rational subset U ⊂ X , we have f M ( U ) = O X ( U ) ⊗ A M ;[Hub94, Theorem 2.5] guarantees that this assignment is indeed a sheaf. UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 47
Definition B.5.1. An O X -module F on a locally strongly noetherian analytic adic space X is coherent if there is an open covering X = ∪ i ∈ I U i by strongly noetherian Tate affinoids such that F | U i ∼ = f M i for a finite O X ( U i )-module M i . Theorem B.5.2.
Let X = Spa( A, A + ) be a strongly noetherian Tate affinoid, and F a coherent O X -module. Then(1) there is a unique finite A -module M such that F ∼ = f M .(2) H i ( X, F ) = 0 for i ≥ Proof. (1) is shown in [Hub93b, p. 3.6.20], and (2) is shown in [Hub94, Theorem 2.5]. (cid:3)
Corollary B.5.3.
Let f : X → Y be a finite morphism of locally strongly noetherian adic spaces.Then(1) coherent O Y -modules are closed under kernels, cokernels, and extensions in Mod O Y ;(2) for any coherent O X -module F , f ∗ F is a coherent O Y -module. Proof.
It suffices to prove the claim under the additional assumption that Y is a strongly noetherianTate affinoid. Now both parts easily follow from Theorem B.2.19, Theorem B.5.2 and flatness of O Y ( Y ) → O Y ( U ) for a rational subdomain U ⊂ Y [Hub94, (II.1), (iv) on page 530]. (cid:3) B.6.
Closed Immersions.
In this section we discuss the notion of closed immersion in the contextof locally strongly noetherian adic spaces.
Definition B.6.1.
We say that a morphism f : X → Y of analytic adic spaces is an open immersion if f is a homeomorphism of X onto an open subset of Y , and the map f − O Y → O X is anisomorphism. Remark B.6.2.
Remark B.1.2 ensures that f : X → Y is an open immersion if and only if f is anisomorphism onto an open adic subspace of Y . Definition B.6.3.
We say that a morphism f : X → Y of locally strongly noetherian analytic adicspaces is a closed immersion if f is a homeomorphism of X onto a closed subset of Y , the map O Y → f ∗ O X is surjective, and the kernel I := ker( O Y → f ∗ O X ) is coherent. Remark B.6.4. If i : X → Y is a closed immersion of (locally strongly noetherian) adic spaceswith I = ker( O Y → i ∗ O X ). Then there is a set-theoretic identification: | X | = { y ∈ Y | ( i ∗ O X ) y ≃ } = { y ∈ Y | I y ≃ O Y,y } . Lemma B.6.5.
Let Y = Spa( A, A + ) be a strongly noetherian Tate affinoid, and i : X → Y a closedimmersion. Then B := O X ( X ) is a complete Tate ring, and the natural morphism i ∗ : A → B is atopological quotient morphism. Proof.
The natural morphism A → B is clearly continuous as O Y → i ∗ O X is a morphism of sheavesof topological rings. Therefore, any topologically nilpotent unit ̟ A ∈ A defines a topologicallynilpotent unit ̟ := i ∗ ( ̟ A ) ∈ B .We now show that B is a Tate ring. Since X is closed in an affinoid, we conclude that X isquasi-compact and quasi-separated. So we choose a finite covering X = ∪ ni =1 U i by open affinoid U i = Spa( B i , B + i ). Then B ⊂ n Y i =1 B i and the topology on B coincides with the subspace topology. Each B i admits a ring of definition B i, , and we can assume that the topology on every B i, is the ̟ -adic topology (possibly afterreplacing ̟ A with a power). We claim that B := n Y i =1 B i, ! ∩ B = n Y i =1 ( B i, ∩ B )is a ring of definition in B . It suffices to show the topology on B induced from Q ni =1 B i, coincideswith the ̟ -adic topology. This follows from the equalities ̟ n ( n Y i =1 B i, ) ∩ B ! = ̟ n n Y i =1 B i, ! ∩ B that, in turn, follow from the fact that ̟ is invertible in B .Now we address completeness of B . By a similar reason, we see that there is a short exactsequence 0 → B d −→ n Y i =1 B i a −→ Y i,j O X ( U i ∩ U j )such that d is a topological embedding and a is continuous. Using that X is quasi-separated, wecan cover each U i ∩ U j by a finite number of affinoids V i,j,k . Thus, we get a short exact sequence0 → B d −→ n Y i =1 B i b −→ Y i,j,k O X ( V i,j,k )such that d is a topological embedding and b is continuous. Every B i = O X ( U i ) and O X ( V i,j,k )is complete by construction. Therefore, we conclude that B is closed inside a complete Tate ring Q ni =1 B i . Thus, it is also complete.Finally, we show that A → B is a topological quotient morphism. We consider a short exactsequence 0 → I → O Y → i ∗ O X → . Theorem B.5.2(2) ensures that H ( Y, I ) = 0, so A → B is surjective. Now A → B is a surjectivecontinuous morphism of complete Tate rings, so it is open by Remark B.2.3. In particular, it is atopological quotient morphism. (cid:3) Lemma B.6.6.
Let (
A, A + ) be a strongly noetherian Tate-Huber pair, and let I be an ideal in A .We define ( A + /I ∩ A + ) c to be the integral closure of A + /I ∩ A + in A/I . Then (
A/I, ( A + /I ∩ A + ) c )is a complete strongly noetherian Tate-Huber pair, and the morphism Spa( A/I, ( A + /I ∩ A + ) c ) → Spa(
A, A + ) is a closed immersion Proof.
First of all, we note that
A/I is complete by [Hub94, Proposition 2.4(ii)] and the naturalmorphism p : A → A/I is open. Now we show that (
A/I, ( A + /I ∩ A + ) c ) is also a Tate-Huber pair.We choose a pair of definition ( A , ̟ ) with ̟ being a pseudo-uniformizer in A . Then openness of p implies that p ( A ) is open in A/I . Moreover, its quotient topology coincides with the p ( ̟ )-adictopology, so it is a ring of definition in A/I . Also, p ( ̟ ) is a topologically nilpotent unit in A/I , so
A/I is a Tate ring. A similar argument shows that ( A + /I ∩ A + ) c is an open subring of A/I thatis contained in (
A/I ) ◦ .We claim that A/I is strongly noetherian. It suffices to show that A h T , . . . , T n i → ( A/I ) h T , . . . , T n i UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 49 is surjective for each n ≥
1. An inductive argument shows that it suffices to prove the claim for n = 1. We pick an element f ∈ ( A/I ) h T i ; it can be written as f = P i a i T i for some a i ∈ A suchthat { a i } is a null-system in A/I . This means that for any m there is N m such that: a i ∈ p ( ̟ ) m p ( A )for any i ≥ N m . Thus we can we can find a sequence ( b i ) of elements of A such that b i = a i forany i ≥ b i ∈ ̟ m A for any i ≥ N m . This means that P i b i T i lies in A h T i and its image in( A/I ) h T i coincides with f .Now we check that the natural morphism i : X := Spa( A/I, ( A + /I ∩ A + ) c ) → Y := Spa( A, A + )is a closed immersion. Firstly, we note that topologically we have an equality i ( X ) = V ( I ) := { x ∈ Spa(
A, A + ) | v x ( I ) = 0 } with v x being the valuation corresponding to a point x . We show that this set is closed. First ofall, we note that it suffices to show that the set V ( f ) := { x ∈ Spa(
A, A + ) | v x ( f ) = 0 } is closed for any f ∈ I since V ( I ) = ∩ f ∈ I V ( f ). And V ( f ) is closed as its complement is equal tothe union of the rational subdomains: Y \ V ( f ) = [ n ∈ N Y (cid:18) ̟ n f (cid:19) We also need to check that the map O Y → i ∗ O X is surjective with coherent kernel. Clearly, i : X = Spa( A/I, A + / ( I ∩ A + ) c ) → Y = Spa( A, A + ) is finite, so i ∗ O X is a coherent O Y -module byCorollary B.5.3(2). Thus, Corollary B.5.3(1) ensures that ker( O Y → i ∗ O X ) is coherent.Now we show that O Y → i ∗ O X is surjective. It suffices to show that for any rational subdomain U = Y (cid:16) f g , . . . , f n g (cid:17) the morphism O Y ( U ) → ( i ∗ O X )( U ) is surjective. This boils down to showingthat the map A (cid:28) f g , . . . , f n g (cid:29) → ( A/I ) (cid:28) f g , . . . , f n g (cid:29) is surjective. Consider the commutative diagram A h T , . . . , T n i ( A/I ) h T , . . . , T n i A D f g , . . . , f n g E ( A/I ) D f g , . . . , f n g E where the upper horizontal arrow is surjective by the discussion above. This implies that the lowerhorizontal arrow is surjective as well. (cid:3) Lemma B.6.7.
Let f : Spa( B, B + ) → Spa(
A, A + ) be a morphism of strongly noetherian Tateaffinoids, and I ⊂ A an ideal. Then the natural morphismSpa( B/IB, ( B + / ( B + ∩ IB )) c ) → Spa(
B, B + ) × Spa(
A,A + ) Spa(
A/I, A + / ( I ∩ A + ) c ) . is an isomorphism. Proof.
Lemma B.3.6 applied to the finite morphism A → A/I ensures thatSpa (cid:0)
B, B + (cid:1) × Spa(
A,A + ) Spa (cid:0) ( A/I ) , A + / ( I ∩ A + ) c (cid:1) ≃ Spa (cid:0) ( A/I ) ⊗ A B, ( A/I ⊗ A B ) + (cid:1) . Now Lemma B.3.4 and [Hub93b, Lemma 2.4(ii)] ensure that the canonical algebraic isomorphism(
A/I ) ⊗ A B ≃ B/IB preserves topologies on both sides. Now we recall that((
A/I ) ⊗ A B ) + = Im (cid:0)(cid:0) A + /I ∩ A + (cid:1) c ⊗ A + B + → ( A/I ) ⊗ A B (cid:1) c = Im (cid:0) B + / ( I ∩ A + ) B + → B/IB (cid:1) c . This admits a natural morphismIm (cid:0) B + / ( I ∩ A + ) B + → B/IB (cid:1) c → ( B + / ( B + ∩ IB )) c that is both injective and surjective. This implies thatSpa (cid:0) B, B + (cid:1) × Spa(
A,A + ) Spa (cid:0) ( A/I ) , A + / ( I ∩ A + ) c (cid:1) ≃ Spa (cid:0) ( A/I ) ⊗ A B, ( A/I ⊗ A B ) + (cid:1) ≃ Spa(
B/IB, ( B + / ( B + ∩ IB )) c ) . (cid:3) Corollary B.6.8.
Let Y = Spa( A, A + ) is a strongly noetherian Tate affinoid, I ⊂ A an ideal, and X = Spa( A/I, ( A + /I ∩ A + ) c ). Then the natural map e I → ker( O Y → i ∗ O X )is an isomorphism. Proof.
This follows from the fact that the formation of Spa(
A/I, ( A + /I ∩ A + ) c ) commutes withbase change by Lemma B.6.7, and the fact that I O Y ( U ) = I ⊗ A O Y ( U ) by A -flatness of O Y ( U ) fora rational subdomain U ⊂ Y . (cid:3) Corollary B.6.9.
Let Y = Spa( A, A + ) is a strongly noetherian Tate affinoid, and let i : X → Y isa closed immersion. Then it is isomorphic to the closed immersion from Spa( A/I, ( A + /I ∩ A + ) c )for a unique ideal I ⊂ A . Proof.
Uniqueness of I is easy. Corollary B.6.8 implies that, for a closed immersion X = Spa( A/I, ( A + /I ∩ A + ) c ) → Y = Spa( A, A + ) , we can recover I as Γ( Y, I ) for I = ker( O Y → i ∗ O X ).Now we show existence of I . We consider a short exact sequence0 → I → O Y → i ∗ O X → . Theorem B.5.2(1) implies that I ∼ = e I for an ideal I ⊂ A , so Lemma B.6.5 ensures that O X ( X ) is acomplete Tate ring and O X ( X ) ≃ A/I topologically. This isomorphism induces a natural morphism φ : X → Spa(
A/I, ( A + /I ∩ A + ) c ) by Remark B.1.5.We firstly show that φ is a homeomorphism. Since both X and Spa( A, ( A + /I ∩ A + ) c ) are topolog-ically closed subsets of Spa( A, A + ), it is sufficient to show that φ is a bijection. Now Remark B.6.4and Corollary B.6.8 imply that both X and Spa( A/I, ( A + /I ∩ A + ) c ) can be topologically identifiedwith the set { y ∈ Y | J y ≃ O Y,y } . Now we use Remark B.1.2 to ensure that it suffices to show that φ : O Spa(
A/I, ( A + /I ∩ A + ) c ) → φ ∗ O X is an isomorphism of sheaves of topological rings. Since i ′ : Spa( A/I, ( A + /I ∩ A + ) c ) → Spa(
A, A + )is topologically a closed immersion, it suffices to show that φ is an isomorphism after applying i ′∗ ,i.e. it suffices to show that the natural morphism i ′∗ O Spa(
A/I, ( A + /I ∩ A + ) c ) → i ∗ O X UOTIENTS OF ADMISSIBLE FORMAL SCHEMES AND ADIC SPACES BY FINITE GROUPS 51 is an isomorphism of sheaves of topological rings. Corollary B.6.8 implies that this is an algebraicisomorphism. We use Remark B.2.3 and Lemma B.6.5 to handle the topological aspect of theisomorphism. (cid:3) Corollary B.6.10.
Let i : X → Y be a closed immersion of locally strongly noetherian adic spaces.Then(1) for any locally topologically finite type morphism Z → Y , the fiber product Z × Y X → X → X is a closed immersion,(2) for any closed immersion i ′ : Z → X , the composition i ◦ i ′ : Z → Y is a closed immersion. Proof.
For the purpose of proving (1), it suffices to assume that X , Y and Z are strongly noetherianTate affinoids. Then the result follows from Lemma B.6.7 and Corollary B.6.9.Similarly to prove (2), it is sufficient to assume that Y is a strongly noetherian Tate affinoid.Then the same holds for X and Z by Corollary B.6.9. It is clear that Z → Y is a homeomorphismonto its closed image, and that O Y → ( i ◦ i ′ ) ∗ O Z is surjective. Thus, we only need to show that itis the kernel of that map is coherent. It suffices to show that ( i ◦ i ′ ) ∗ O Z is coherent. Now we notethat i and i ′ are finite by Corollary B.6.9, so i ◦ i ′ is also finite. Therefore, ( i ◦ i ′ ) ∗ O Z is coherentby Corollary B.5.3(2). (cid:3) Definition B.6.11.
We say that a morphism f : X → Y of analytic adic spaces is a locally closedimmersion if f can be factored as j ◦ i where i is a closed immersion and j is an open immersion. Lemma B.6.12.
Let f : X → Y and g : Y → Z be locally closed immersions of locally stronglynoetherian adic spaces. Then so is g ◦ f . Proof.
We firstly deal with the case f an open immersion and g a closed immersion. In thiscase the topology on Y is induced from Z , so there is an open adic subspace U ⊂ Z such that X = U ∩ Z = g − ( U ). Therefore, we can factor g ◦ f as X a −→ U b −→ Z. We note that a is a closed immersion as the restriction of the closed immersion g over U ⊂ Z , and b is an open immersion by construction. Hence, g ◦ f is indeed an immersion.Now we consider the general case. In this case we can factor f as j ◦ i with a closed immersion i and an open immersion j . Similarly, we can factor g = j ′ ◦ i ′ with a closed immersion j ′ and anopen immersion i ′ . The argument above implies that the composition i ′ ◦ j can be rewritten as j ′′ ◦ i ′′ for a closed immersion immersion i ′′ and an open immersion j ′′ . Therefore, g ◦ f = j ′ ◦ i ′ ◦ j ◦ i = j ′ ◦ j ′′ ◦ i ′′ ◦ i = ( j ′ ◦ j ′′ ) ◦ ( i ′′ ◦ i ′ ) . Now i ′′ ◦ i ′ is a closed immersion by Corollary B.6.10(2), and clearly j ′ ◦ j ′′ is an open immersion.Therefore, g ◦ f is an immersion. (cid:3) Remark B.6.13.
The order of an open and a closed immersion in Definition B.6.11 is needed toensure that a composition of immersions is an immersion; the same happens over C . Lemma B.6.14.
Let f : X → Y be a locally closed immersion of analytic adic spaces such thatthe image f ( X ) is closed in Y . Then f is a closed immersion. And an obvious observation that restriction of a closed immersion over an open subspace of the target is againa closed immersion.
Proof.
We write f as a composition X i −→ U j −→ Y of a closed immersion i and an open immersion j . Since both i and j are topological embeddings,the same holds for f . Moreover, its image is closed in Y by hypothesis on f . So we are left to showthat I := ker( O Y → f ∗ O X ) is coherent, and f : O Y → f ∗ O X is surjective.We use an open covering Y = U ∪ ( Y \ f ( X )). We know that I | U is coherent by assumption, andit is clear that I | Y \ f ( X ) ≃ I is coherent on Y .Now we show surjectivity of f ♯ . We note that since f is topologically a closed embedding, weconclude that ( f ∗ O X ) y ∼ = 0 for any y / ∈ f ( X ). So it suffices to check surjectivity on stalks for y ∈ f ( X ) ⊂ U . But then f y is identified with O Y,y ∼ = O U,y ։ O X,y by the assumptions on i and j . (cid:3) B.7.
Separated Morphisms of Adic Spaces.Definition B.7.1.
We say that a locally topologically finite morphism f : X → Y of locally stronglynoetherian analytic adic spaces is separated , if the diagonal morphism ∆ X/Y : X → X × Y X hasclosed image Remark B.7.2.
We assume that f is locally topologically finite type to ensure the existence ofthe fiber product X × Y X . Lemma B.7.3.
Let f : X → Y be a locally topologically finite morphism of locally stronglynoetherian analytic adic spaces. Then ∆ X/Y : X → X × Y X is a locally closed immersion. Proof.
We cover Y by strongly noetherian Tate affinoids ( U i ) i ∈ I , and then we cover the pre-images f − ( U i ) by strongly noetherian Tate affinoids ( V i,j ) j ∈ J i . The construction of fiber products in[Hub96, Proposition 1.2.2 (a)] implies that ∪ i,j V i,j × U i V i,j is an open subset in X × Y X thatcontains ∆ X ( X ). Thus in order to show that ∆ X/Y is an immersion, it is suffices to show α : X → ∪ i,j V i,j × U i V i,j is a closed immersion.Moreover, we note that α − ( V i,j × V i,j ) = V i,j for any i ∈ I, j ∈ J i . Since the notion of a closedimmersion is easily seen to be local on the target, we conclude that it is enough to show that thediagonal morphism is a closed immersion for affinoid spaces X = Spa( B, B + ) and Y = Spa( A, A + ).But then the diagonal morphism X → X × Y X coincides with the morphism∆ X/Y : Spa(
B, B + ) → Spa (cid:16) B b ⊗ A B, (cid:0) B b ⊗ A B (cid:1) + (cid:17) induces by the natural “multiplication morphism” of Tate-Huber pairs m : (cid:16) B b ⊗ A B, (cid:0) B b ⊗ A B (cid:1) + (cid:17) → ( B, B + )with (cid:0) B b ⊗ A B (cid:1) + being the integral closure of B + b ⊗ A + B + inside B b ⊗ A B . Then we see that ∆ X/Y isa closed immersion by Lemma B.6.6. (cid:3)
Corollary B.7.4.
Let f : X → Y be a locally topologically finite type, separated morphism oflocally strongly noetherian analytic adic spaces. Then the diagonal morphism ∆ X/Y : X → X × Y X is a closed immersion. Proof.
This follows from Lemma B.6.14 and Lemma B.7.3. (cid:3)
EFERENCES 53
Corollary B.7.5.
Let f : X → S be a locally topologically finite type, separated morphism ofanalytic adic spaces. Suppose that S = Spa( A, A + ) is a strongly noetherian Tate affinoid, and that U and V are two open affinoids in X . Then their intersection U ∩ V is also an open affinoid in X . Proof.
Consider the following commutative diagram: U ∩ V U × S VX X × S X i ∆ X/S
Since the map ∆
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