Quotients Of Admissible Formal Schemes and Adic Spaces by Finite Groups

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 strongly noetherian adic space.

1. Introduction 1.1. 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 nonnoetherian valuation ring, admissible formal schemes over a complete microbial valuation ring (see Definition 3.1.1), and locally topologically finite type adic spaces over a locally strongly noetherian analytic adic space. These 3 different contexts occupy Chapters 2, 3, and 4 respectively.
The motivation to study these quotients comes from our paper [Zav21a], where we show a refined version of Temkin's local alteration theorem. Our result roughly says that any smooth rigid-space X over an algebraically closed non-archimedean field C locally admits a formal O C -model X such that X is a quotient of a polystable admissible formal O C -model X ′ by a finite group G (acting freely on the generic fiber). This refined uniformization result is an important technical input in the author's proof of p-adic Poincaré Duality in [Zav21b].
The actual formulation of this uniformization result is quite technical, and we refer to [Zav21a,Theorem 1.4] for the precise formulation. We only mention that, in order to formulate and prove this theorem, we had to make sure that a quotient of an admissible formal O C -scheme by an O Caction 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 never noetherian.
1.1.1. Scheme Case. Before we deal with quotients of formal schemes and adic spaces, we first discuss quotients of schemes over the base scheme S = Spec O C . Even this question is already nontrivial and demonstrates an important source of difficulties in the question of studying quotient spaces in the non-noetherian situation. The same difficulty will arise in every other setup treated in this paper.
We fix an S-scheme X with an S-action of a finite group G. Then a standard argument constructs the quotient X/G as an S-scheme (under some assumptions); this is carried out in [SGA I, Exp. V, §1] (see also Definition 2.1.1 and Theorem 2.1.15). However, the question of whether, for a finite type S-scheme X, the quotient X/G is of finite type is quite non-trivial.
To explain the main issue, we briefly recall what happens in the classical situation of a finite type R-scheme X with an R-action of a finite group G for some noetherian ring R. Under some mild assumptions 1 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: one firstly 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 is finitely generated over R.
One may think that probably the Artin-Tate lemma can hold, more generally, over a nonnoetherian base R if C is finitely presented over R. However, this is not the case and the Artin-Tate Lemma fails over any non-noetherian base: 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 2 ), and the R-subalgebra B = R ⊕ Iε. So C is a finitely presented R-algebra, and C is finite as a 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-noetherian situations 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 features of 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 work in 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 3.2.5 and Lemma 4.2.4). However, there are issues that are not seen in the scheme case. We explain a few of the main new technical difficulties that arise while proving the result in the world of adic spaces.
Compared to the affine (formal) schemes, the underlying topological space of an affinoid space Spa(A, A + ) is harder to express in terms of the pair (A, A + ). It is a set of all valuations on A with corresponding continuity and integrality conditions. 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 geometric meaning. Hence, it takes extra care to identify Spa(A G , A +,G ) with Spa(A, A + )/G even on the level of underlying topological spaces.
Furthermore, the notion of a topologically finite type (resp. finite) morphism of Tate-Huber pairs is more subtle than its counterpart in the algebraic setup for two different reasons. Firstly, it has a topological aspect that takes some care to work with. Secondly, the notion involves conditions on both A and A + (see Definition B.2.1 and Definition B.2.6). Usually, A + is non-noetherian, so it requires 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 finite type over a strongly noetherian analytic adic space in Section 4. One reason for this level of generality is to include adic spaces that are topologically finite type over Spa(k, k + ) for a microbial valuation ring k + (see Definition 3.1.1). These spaces naturally arise while studying fibers of morphisms of rigid spaces 2 X → Y over points of Y of higher rank. We think that the category of strongly noetherian analytic adic spaces is the natural one to consider 3 . One of its advantages is that it contains both topologically finite type morphisms and morphisms coming from the (not necessary finite) base field extension in rigid geometry.
In the case of formal schemes, the results of Section 3 are written in the generality of admissible formal schemes over a complete, microbial valuation ring k + (see Definition 3.1.1). We want to point out that Appendix A contains versions of the main results of Section 3 for a topologically universally adhesive base (see Definition A.3.12). These results are more general and include both the cases of formal schemes topologically finite type (and flat) over some k + and noetherian formal schemes. However, we prefer to formulate and prove the results in the main body of the paper for admissible formal schemes over k + since it simplifies the exposition a lot. We only refer to Appendix A for the necessary changes that have to be made to make the arguments work in the more general adhesive situation.
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 adhesive only if it is microbial (see Lemma A.2.3), so the results of Appendix A do not fully subsume the results of Section 2.2. [Han21]. While writing this paper, we found that similar results for adic spaces were already obtained in [Han21]. We briefly discuss the main similarities and differences in our approaches. David Hansen separately discusses two different situations: rigid spaces 4 over a non-archimedean field K, and general analytic adic spaces. In the former case, he shows that (under some assumptions on 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 as an analytic adic space if the order of G is invertible in O X (X). The argument there is based on an averaging trick, so it uses the invertibility assumption in order to be able to divide by #G. We note that if X is a perfectoid space over a perfectoid field, he can drop this invertibility assumption 2 Considered as adic spaces 3 It may be also reasonable to consider some class of non-analytic adic spaces. However, it is not clear what should be the correct uniform condition on an adic space that would imply the Artin-Tate lemma in both analytic and nonanalytic non-noetherian setups. Since we never need to use non-analytic adic spaces in our intended applications, we prefer to work only with analytic adic spaces in this paper. 4 Defined as locally topologically finite type adic spaces over Spa (K, OK ) by some other argument. The whole point of the latter case is to be able to work with "big" adic spaces such as perfectoid spaces.

Comparison with
In contrast with Hansen's approach, our methods neither use any non-trivial input from nonarchimedean analysis, nor the averaging trick. What we do is try to imitate the classical algebraic argument based on the Artin-Tate Lemma in the setup of strongly noetherian adic spaces. More precisely, we show that if X is a locally topologically finite type adic space (with some other conditions) 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 not recover Hansen's result as we do not allow "big" adic spaces such as perfectoid spaces, but it proves a stronger statement in the case of adic spaces locally of finite type over a locally strongly noetherian S as we do not have any assumptions on the order of G. Moreover, even in the case of rigid spaces, it gives a new proof of the existence of X/G as a rigid space that does not use much of analytic theory.
1.3. Our results. We firstly study the case of a flat, locally finite type scheme X over a valuation ring 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 2.1.15 and 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 X/G exists as a flat, locally finite type k + -scheme. Moreover, it satisfies the following properties: (1) π : X → X/G is universal in the category of G-invariant morphisms to locally ringed Sspaces.
(3) Fibers of π are exactly the G-orbits.
(4) The formation of the quotient X/G commutes with flat base change (see Theorem 2.1.15 (4) for the precise statement).
We then consider quotients of admissible formal schemes X over a complete microbial valuation ring k + by a k + -action of a finite group G. Under similar conditions, we show that X/G exists as an 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 a finite 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 locally ringed spaces over S.
(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 strongly noetherian adic space. Theorem 1.3.3. (Theorem 4.3.4) Let S be a locally strongly noetherian analytic adic space (see Definition B.2.15), and X a locally topologically finite type adic S-space with an S-action of a finite group G. Suppose that each point x ∈ X admits an affinoid open neighborhood V x containing G.x. Then X/G exists as a locally topologically finite type adic S-space. Moreover, it satisfies the following properties: (1) π : X → X/G is universal in the category of G-invariant morphisms to topologically locally v-ringed S-spaces (see Definition B.1.1).
(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 condition that each point x ∈ X admits an affinoid open neighborhood V x containing G.x is a much milder in the adic world than in the scheme world. For example, we show that it is automatic if X is a separated rigid-analytic space. This, in particular, implies that a quotient of a separated rigid-analytic space by a finite group action always exists as a rigid-analytic space. In case of a free finite group action, a similar result has been previously obtained in [CT09, Theorem 5.1.1] in the world of Berkovich spaces. Lemma 1.3.4. (Lemma 4.3.6) Let K be a non-archimedean field with the residue field k, X a separated, locally finite type adic Spa(K, O K )-space, and {x 1 , . . . , x n } is a finite set of points of X. Then there is an open affinoid subset U ⊂ X containing all x i . Remark 1.3.5. It is reasonable to expect that the assumption of Theorem 1.3.3 is automatic as long as X is S-separated (see Remark 4.3.5). However, the proof of this claim would seem to require a generalization of the main results of [Tem00] to more general adic spaces. This is beyond the scope of this paper.
The natural question is whether these quotients commute with certain functors like formal completion, analytification, and adic generic fiber. We show that this is indeed the case, i.e. the formation of the geometric quotients commutes with the functors mentioned above whenever they are defined. We informally summarize the results below: (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. The natural morphism X/G → X/G is an isomorphism.
(2) Let k + be a complete, microbial valuation ring with fraction field k, and X an admissible formal k + -scheme with a k + -action of a finite group G. Suppose X satisfies the assumption of 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 2.1.15. The natural morphism X an /G → (X/G) an is an isomorphism.

Quotients of Schemes
2.1. Review of Classical Theory. We review the classical theory of quotient of schemes by an action of a finite group. This theory was developed in [SGA I, Exp. V, §1]. We review the main results from there, and present some proofs in a way that will be useful for our later purposes. This section is mostly expository.
For the rest of this section, we fix a base scheme 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 unique morphism induced by f : X → S and h # is the natural morphism We note that X/G is, a priori, only a ringed space. In the lemma below, we show that it is actually always a locally ringed space: 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 2.1.3. 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 2.1.2 allows us to construct quotients entirely in the category of locally ringed spaces and not merely in the category of all ringed spaces. The main technical issue with the category of ringed spaces is that locally ringed spaces do not form a full subcategory of it.
Proof of Lemma 2.1.2. We note that the action of G induces a family of ring isomorphisms is commutative. In particular, G acts on O X (U ) for any G-stable open U ⊂ X. We describe the stalk O X/G,x for a point x ∈ X/G with a lift x ∈ X as follows: That being said, we wish to show that is a local homomorphism of local rings. This is equivalent to saying that 5 m x ∩ O X/G,x is the unique ). Uniqueness of the inverse element and the sheaf axioms imply that Remark 2.1.4. It is straightforward 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 2.1.5. 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/2Z. Namely, Hironaka's example [Ols16, Example 5.3.2] is a smooth and proper, connected 3-fold over C with a C-action of Z/2Z such that there is an orbit G.x that is not contained in any open affine subscheme U ⊂ X. Lemma 2.1.7 below implies that X/G is not a scheme. only a (rather straightforward) statement that it is impossible for X/G to be a scheme and for π : X → X/G to be affine 7 . We strengthen the result and show that X/G is not a scheme without the affineness requirement on π.
5 In what follows, we slightly abuse the notation and write T ∩ O X/G,x as an abbreviation for π ♯ x −1 for any subset T ⊂ OX,x. 6 To use this result, we recall that the structure sheaf OY is ample on any affine scheme Y . 7 Affineness of X → X/G is part of the definition of an "admissible" action of G on X introduced in [SGA I, Exp.
Proof. Suppose that X/G is an R-scheme, and consider the image x := π(x) ∈ X/G. It admits an affine neighborhood U ⊂ X/G; this defines an open G-stable subscheme U := π −1 (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 separated of finite type since U is separated of finite type over R and U is separated; its fibers are finite by construction. Therefore, Zariski's main theorem [EGA IV 4 , Proposition 18.12.12] implies that π| U is quasi-affine, i.e. the natural morphism is a quasi-compact open immersion. We note that Spec O U (U ) naturally admits an action of the group G induced by the action of G on O U . Trivially, any point y ∈ Spec O U (U ) admits an affine neighborhood containing G.y. Thus, Lemma 2.1.6 applied to Spec O U (U ) and its open subscheme U implies that the same holds for U . As a result, the orbit G.x is contained in some open affine subscheme of X.
Definition 2.1.1 is useless unless we can verify that X/G is a scheme if X is. The main goal of the rest of the section is to review when this is the case under some (mild) assumptions on X.
We start with the case of an affine scheme X = Spec A and an affine scheme S = Spec R. Then the 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 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 2.1.9. 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 I, Expose V, Proposition 1.1(i), (ii) and Corollaire 1.5]. We also point out that the results follow from [AM69, Exercise 5.12, 5.13], and the observation that an integral, finite type morphism is finite.
Proof. This is shown in [SGA I, Exp. V, Proposition 1.1(iv)]. We review this argument here as this type of reasoning will be adapted to more sophisticated situtions later in the paper.
Step 1. φ is a homeomorphism: We note that Lemma 2.1.9 ensures that p : X → Spec A G is a closed, 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 induced map f is clearly a homeomorphism.
Step 2. φ is an isomorphism of locally ringed spaces: We use Lemma 2.1.9 again to check that the morphism of sheaves φ # : O Y → φ * O X/G is an isomorphism. Using the base of basic affine opens in Y , it suffices to show that the map Now we want to discuss when X/G exists as a scheme in the global set-up without a separatedness assumption. Roughly, we want to cover X by G-stable affines and then deduce the claim from Proposition 2.1.12. In order to do this, we need the following lemma: Lemma 2.1.13. Let X be an S-scheme with an S-action of a finite group G. Suppose that for any point x ∈ X there is an open affine subscheme V x ⊂ X that contains the orbit G.x. Then each point x ∈ X has a G-stable open affine neighborhood U x ⊂ X.
Proof. The proof is outlined just after [SGA I, Exp V, Proposition 1.8], we recall the key steps here. Firstly, Lemma 2.1.6 ensures that one can reduce to the case of an affine base S = Spec R. Then one shows the claim for a separated X, in which case U x := g∈G g(V x ) is affine and does the job. In general, Lemma 2.1.6 guarantees that one can replace X with the separated open subscheme g∈G g(V x ) and reduce to the separated case. We recall one case where the condition of Lemma 2.1.13 is satisfied. Proposition 2.1.14. Let φ : X → S be a locally quasi-projective 8 S-scheme with an S-action of a finite 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 there is a quasi-compact R-immersion X ⊂ P N R . Then it suffices to show a stronger claim that any finite set of points is contained in an open affine. This is shown in [EGA II, Corollaire 4.5.4]. Now, we are ready to explain the main existence result [SGA I, Exp V, Proposition 1.8]. For later needs, we give a slightly different proof.
Theorem 2.1.15. Let X be an S-scheme with an S-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 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 Sspaces.
(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 flat morphism Z → X/G, the geometric quotient (X × X/G Z)/G is a scheme, and the natural morphism (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 2.1.6 to reduce to the case where S is affine. Now Lemma 2.1.13 allows to cover X by G-stable open affine subschemes U i . Then the construction of the geometric quotient implies that is an open subset that is naturally isomorphic to U i /G, and π −1 (U i /G) coincides with U i . This implies that it suffices to show that U i /G is a scheme. This was already shown in Proposition 2.1.12.
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 2.1.9.
Step 3. π : X → X/G is universal and commutes with flat base change: The universality is essentially trivial (Remark 2.1.4). 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 from Lemma 2.1.11 and the identification of X/G with Spec A G .

2.2.
Schemes Over a Valuation Ring k + . The main drawback of Theorem 2.1.15 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 makes this theorem not so useful in practice as we often do not want to leave the realm of finite type morphisms. The main work left is to show that the ring of invariants A G is finite type over R if A is. If R is noetherian, this problem is resolved using the Artin-Tate Lemma 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 .  (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.
Proof. By [Mat86, Theorem 7.7] a k + -module N 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 (see also [Sta21, Tag 0539] for a different proof).
The second and third claims are proven in [Sta21, Tag 053E]. Now we show the last claim. We consider the quotient module M/N . The saturatedness assumption 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 homomorphism from a finite module to a finitely presented one (see [Sta21, Tag 0519]).
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 (in the sense of Definition 2.2.1). Then A is a k + -algebra of finite type.
Proof. By 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 1 , . . . , T n ] → 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 1 , . . . , y m ∈ B. The choice of x 1 , . . . , x n and y 1 , . . . , y m implies that there are some a i,j , a i,j,l ∈ A with the relations Now consider the k + -subalgebra A ′ of A generated by all a i,j and a i,j,l . Clearly, A ′ is of finite type over k + . Moreover, B is finite over A ′ as y 1 , . . . , 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 ′ -submodule of the finite A ′ -module B. Therefore, A is of finite type over k + .
Corollary 2.2.4. Let A be a flat, finite type k + -algebra with a k + -action of a finite group G. Then A G is a finite type flat k + -algebra, and the natural morphism A G → A is finitely presented.
Proof. Lemma 2.1.9 gives that A is a finite A G -module, and A G is easily seen to be saturated in A using that A is k + -torsionfree (because it is k + -flat). Therefore, Lemma 2.2.2 (1) implies that A G is k + -flat and Lemma 2.2.3 ensures that A G is finite type over k + . Now Lemma 2.2.2 (2) guarantees 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 [ 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 the scheme X/G as in Theorem 2.1.15 is flat and locally finite type over k + , and the integral surjection π : X → X/G is finite and finitely presented.
Proof. By construction, X/G is clearly k + -flat. To show that X/G is locally of finite type and that π is finitely presented, we reduce to the affine case by passing to a G-stable affine open covering of X (see Lemma 2.1.13

Quotients of Admissible Formal Schemes
We discuss the existence of quotients for some class of formal schemes by an action of a finite group G. The strategy to construct the quotient spaces is close to the one used in Section 2. We first 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) that X/G is a formal scheme. We resolve this issue by first showing that it is a formal scheme if X is affine. 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 anymore firstly show that X/G is a formal scheme by a very general argument and then study its properties under further assumptions, e.g. show that it is flat or (topologically) finite type over the base. The problem can be seen even in the case of an affine formal scheme X = Spf A. The proof of Proposition 2.1.12 crucially uses that the localization (A G ) f is A G -flat for any f ∈ A G . The analogue in the world of formal schemes would be that the completed localization (A G ) {f } = (A G ) f is A G -flat. However, this requires some finiteness assumption on A G in order to hold. Therefore, we need to verify algebraic properties of A G at the same time as constructing the isomorphism The second, related problem is that one needs to be more careful with certain topological aspects of the theory. For instance, the fiber product of affine formal schemes is given by the completed tensor product on the level of corresponding algebras. This is a more delicate functor as it is neither left nor right exact. So we pay extra attention to make sure that these complications do not cause any issues under suitable assumptions.
3.1. The Setup and the Candidate Space X/G.   For the rest of the section, we fix a complete, microbial valuation ring k + with a pseudouniformizer ̟. We denote by S the formal spectrum Spf k + .
A formal k + -scheme will always mean a ̟-adic formal k + -scheme. It is easy to see that this notion does not depend on the 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 finite type (i.e. there is a surjection k + t 1 , . . . t d → A).
A formal k + -scheme X is called admissible if it is k + -flat and locally topologically of finite type.
(1) We note that there are many (non-equivalent) ways to define flatness in formal geometry. They are all equivalent for a morphism f : X → Y of locally topologically finite type formal k + -schemes.
We prefer to use the following as the definition: (3) In particular, if X = Spf A is an admissible formal k + -scheme, the k + -algebra A is admissible.
We summarize the main properties of locally topologically finite type formal k + -schemes in the lemma below: Lemma 3.1.6. Let k + be a complete, microbial valuation ring, A 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.  (6) For any element f ∈ A, the completed localization The first five results of this lemma are essentially due to Raynaud and Gruson [RG71]. Proof.
Definition 3.1.7. Let G be a finite group, and X a locally topologically 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 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 unique morphism induced by f : X → S and h # is the natural morphism Remark 3.1.8. By construction, X/G is a topologically ringed S-space, and π : X → X/G is a morphism of topologically ringed S-spaces. Furthermore, Lemma 2.1.2 ensures that X/G is a topologically locally ringed S-space, and π is a morphism of 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 category of G-invariant morphisms to topologically locally ringed S-spaces.
Our main goal is to show that under some mild assumptions, X/G is an admissible formal Sscheme when X is. We start with the case of affine formal schemes and then move to the general case.
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. We point out that in contrast to 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 the section with studying certain properties of the ring of invariants A G .  Proof. Since N is closed in M , it is complete for the subspace topology. All this means by design is that is an isomorphism, and we need to justify that this implies that is an isomorphism. For this purpose, we will crucially use that I is finitely generated. We start by considering the diagram Since γ is an isomorphism, we conclude that α is injective. Therefore, it suffices only to justify that α is surjective. By [Sta21, Tag 090S] (which uses that I is finitely generated), it suffices to justify surjectivity of ensures that M is f -adically complete. Therefore, we may assume that I = (f ) and show that the natural morphism is surjective. Now we use Diagram (3) and the fact that γ is an isomorphism to reduce the question to showing that β : Injectivity of β boils down to showing that, for any sequence of elements {a n ∈ N } with a n+1 − a n ∈ f n N and a n ∈ f n M ∩ N , we have a n ∈ f n N .
The assumption on a n implies that a n = a n+1 + f n x n for some x n ∈ N . We note that the sum For this we observe that the partial sums of are equal a n − a n+m . Since a n+m ∈ f n+m M ∩ N for any m ≥ 1 and N is complete in the subspace topology, we conclude a n = f n (x n + f x n+1 + f 2 x n+2 + . . . ) = f n b n finishing the proof.
Corollary 3.2.3. Let (R, I) be a ring with a finitely generated ideal I, and A an I-adically complete R-algebra with an R-action 9 of a finite group G. Then A G is complete in the I-adic topology.
Therefore, the result follows directly from Lemma 3.2.2.
Lemma 3.2.4. 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 finite as an A G -module.
Proof. The first claim is Corollary 3.2.3. The second claim is clear by k + -flatness of A. Thus we only need to show the last claim.
Lemma 2.1.9(1) guarantees that A G → A is integral. However, the proof of finiteness in Lemma 2.1.9(3) is not applicable here since A is not necessarily finite type over k + : it is only topologically 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 type morphism. Since an integral map of finite type is finite, we conclude that the morphism A G /̟A G → A/̟A is finite. Therefore, the successive approximation argument (or [Sta21, Tag 031D]) implies that A is finite as an A G -module .
Lemma 3.2.5 (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 k +submodule 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 keep track of topological aspects of our algebras in order to work with topologically finite type algebras in a meaningful way.
Proof. Since B is topologically finite type over k + , we can choose a finite set of elements x 1 , . . . , x n such that the natural k + -linear continuous homomorphism Since B is a finite A-module, we can choose some A-module generators y 1 , . . . , y m ∈ B. The choice of x 1 , . . . , x n and y 1 , . . . , y m implies that there are some a i,j , a i,j,l ∈ A and relations We consider the k + -algebra A ′ := k + T i,j , T i,j,l 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 ̟-adically complete.
By definition A ′ is topologically finite type over k + , and we claim that B is finite over A ′ since it is generated by y 1 , . . . , y m as an A ′ -module. To see this we note that it suffices to show it mod ̟ by successive approximation (or [Sta21, 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 + -algebra is also topologically finite type.
Corollary 3.2.6. 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 3.2.4 to see that A G is ̟-adically complete, and A G → A is saturated. Then Lemma 3.2.5 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 induced topology on A G coincides with the ̟-adic topology by Lemma 3.1.6(5).

Now Lemma 2.2.2 (1) implies that
Remark 3.2.7. One can show that the ̟-adic topology on A G coincides with the induced topology from first principles. But we prefer the proof above as it generalizes better to the topologically universally adhesive situation (see Definition A.3.1). Namely, Lemma 3.2.5 and Corollary 3.2.6 hold over any I-adically complete base ring R that is topologically universally adhesive (see Definition A.3.1). We refer to Lemma A.3.6 and Corollary 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 3.2.8. Let X = Spf A be an affine admissible formal k + -scheme with a k + -action of a finite group G. Then the natural map φ : X/G → Y = Spf A G is a k + -isomorphism of topologically locally 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 by Remark 3.1.5(3) (and the analogous fact for topologically finitely generated morphisms). Now the claim immediately follows from Corollary 3.2.6.
Step 1. φ is a homeomorphism: This is completely analogous to Step 1 of Proposition 2.1.12. We only need to show that p : Spf A → Spf A G is a surjective, finite morphism with fibers being exactly G-orbits.
Lemma 3.2.4 says that Spf A → Spf A G is finite. We note that surjectivity of Spec A → Spec A G obtained in Lemma 2.1.9(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. So Lemma 2.1.9(2) ensures that the fibers of Spf A → Spf A G are exactly G-orbits.
Step 2. φ is an isomorphism of topologically locally ringed spaces: We already know that φ is a homeomorphism. So the only thing that we need to show here is that the morphism is an isomorphism of topological sheaves. Using the basis of basic affine opens in Y, it suffices to show that is a topological isomorphism for f ∈ A G . Corollary 3.2.6 ensures that both sides have the ̟-adic topology, so we can ignore the topologies. Now we show that (4) is an (algebraic) isomorphism. We note that 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 is an isomorphism of k + -algebras. This follows from Lemma 2.1.11 and Lemma 3.1.6(6).
Remark 3.2.9. Proposition 3.2.8 can be generalized to the case of an affine, universally adhesive base 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 2.1.15. Proof. This follows easily from Lemma 2.1.6 as |X| ≃ |X × Spf k + Spec k + /̟| and |S| = | Spf k + | ≃ | Spec k + /̟|.
Thus, we can reduce the statement to the case of schemes.
Remark 3.3.3. We note that the condition of Lemma 3.3.2 is automatically satisfied if the special fiber X := X × Spf k + Spec k + /m k is quasi-projective over Spec k + /m k . This follows easily from Proposition 2.1.14.
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 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 locally ringed spaces 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 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 an isomorphism. Proof.
Step 1. The geometric quotient X/G is an admissible formal k + -scheme: The same proof as used in the proof of Theorem 2.1.15 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 3.2.8 to show the claim in the affine case.
Step 2. π : X → X/G is surjective, finite, topologically finitely presented, and fibers are exactly the 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 3.2.4 says that X → X/G is finite. Corollary 3.2.6 ensures that A is finitely presented as an A G -module. Therefore, it is topologically finitely presented as an A G -algebra because [Bos14, Proposition 7.3/10] gives that A G → A is topologically finitely presented if and only if A G /̟ n A G → A/̟ n A is finitely presented for any n ≥ 1.
Step 3. π : X → X/G is universal and commutes with flat base change: The universality is essentially 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 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 the topologically finite type k + -algebra B. Therefore, it suffices to show that the natural map is a topological isomorphism. Both sides have the ̟-adic topology by Corollary 3.2.6. So we can ignore the topologies. Now (5) is an isomorphism by Lemma 2.1.11 and flatness of A G → B (see Remark 3.1.5).
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. The main goal of this section is to compare the schematic and formal quotients by finite groups actions. Throughout this section, we fix a microbial valuation ring k + and a pseudo-uniformizer ̟ ∈ k + . Unlike previous sections, we do not assume that k + is complete.
If X is a flat, locally finite type k + -scheme, we define X to be the formal ̟-adic completion of X. This is easily seen to be an admissible formal k + -scheme with a k + -action of G. Using the universal property of geometric quotients, there is a natural morphism 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 ̟-adic completion X with the induced k + -action of G, and the natural morphism: Step 1. The condition of Theorem 3.3.4 is satisfied for X with the induced action of G: Firstly, we observe that X is k + -admissible as stated above. Now Lemma 2.1.13 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 = ∪ i∈I U i is an open covering of X by affine, G-stable open formal subschemes. In particular, every orbit lies in an affine open formal subscheme of X.
Step 2. We show that X/G → X/G is an isomorphism: We have a commutative diagram of admissible formal 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.
induced by the continuous homomorphism whose source has the ̟-adic topology by construction and whose target has the ̟-adic topology by Corollary 3.2.6. So it suffices to show that this map is an isomorphism of abstract rings (ignoring topology) for any flat, finitely generated k + -flat algebra A. We note that Corollary 2.2.4 shows that A G is a finite type k + -algebra and Lemma 2.1.9 (1) shows that A is a finite A G -module. Therefore, [Bos14,Lemma 7.3/14] implies that the natural homomorphism A ⊗ A G A G → A is an (algebraic) isomorphism. Thus we can identify (6) with the natural homomorphism that is an (algebraic) isomorphism by Lemma 2.1.11 and flatness of the map A G → A G (see [Bos14,Lemma 8 Remark 3.4.2. Theorem 3.4.1 has a version over any topologically universally adhesive base 10 (R, I) (see Definition A.3.1). We refer to Theorem A.3.16 for the precise statement.

Quotients of Strongly Noetherian Adic Spaces
We discuss the existence of quotients of some class of analytic adic spaces by an action of a finite group G. We refer the reader to Appendix B for a review of the main definitions and facts from the 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 firstly construct the candidate space X/G that is, a priori, only a topologically locally v-ringed space (see Definition B.1.1). This construction clearly satisfies the universal property, but it is not clear whether X/G is an adic space. We resolve this issue by firstly showing that it is an adic space if X is affinoid. Then we argue by gluing to prove the claim for a larger class of adic spaces.
We point out the two main complications compared to Section 3 (and Section 2). The first new issue that is not seen in the world of formal schemes is that the notion of a finite (resp. topologically finite type) morphism of Huber pairs (A, A + ) → (B, B + ) is more involved since there is an extra condition on the morphism A + → B + that makes the theory more subtle (see Definition B.2.1 and Definition 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 with corresponding continuity and integrality conditions. So one needs some extra work to identify Spa(A G , A +,G ) with Spa(A, A + )/G even in the affinoid case.
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 11 k(x) → k(x) and the valuation v x : • the morphism h : X/G → S defined by the pair (h, h # ), where h : |X|/G → S is the unique morphism induced by f : X → S and h # is the natural morphism Remark 4.1.3. We note that Lemma 2.1.2 ensures that X/G is a topologically locally v-ringed S-space, and π : X → X/G is a morphism of topologically locally v-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 category of G-invariant morphisms to topologically locally v-ringed S-spaces.
Our main goal is to show that under some assumptions, X/G is a locally topologically finite type adic S-space when X is. We start with the case of affinoid adic spaces and then move to the general case.

Affinoid Case.
For the rest of this section, we assume that S = Spa(R, R + ) is a complete Tate affinoid.
We show that X/G is a topologically finite type adic S-space when X = Spa(A, A + ) for a topologically finite type complete (R, R + )-Tate-Huber pair (A, A + ) with an (R, R + )-action of a finite group G.
We start the section by discussing algebraic properties of the Tate-Huber pair A G , A +,G . In particular, we show that it is topologically of finite type over (R, R + ) if (R, R + ) is strongly noetherian. The main new input is the "analytic" Artin-Tate Lemma 4.2.4. Then we show that the canonical morphism X/G → Spa(A G , A +,G ) is an isomorphism. In particular, X/G is an adic space, topologically of finite type over S.
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 pseudo-uniformizer ̟ ∈ R + and a compatible pair of definition 13 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 0 coincides with the ̟-adic topology, it suffices to show that ̟ n A 0 ∩ A G 0 = ̟ n A G 0 . This can be easily seen from the fact that ̟ is a unit in A (and so a non zero divisor in A 0 ). Now we note that A G 0 is complete in the subspace topology since the action of G on A 0 is clearly continuous. Therefore, it is complete in the ̟-adic topology as these topologies were shown to be equivalent. Also, we note that A G 0 with the subspace topology is clearly open and bounded in A G , so it is a ring of definition by [Hub93b, Proposition 1.1].
Finally, we note that clearly  Proof. Firstly, we note that Lemma 4.2.1 ensures that (A G , A +,G ) is a complete Huber-Tate pair, so it makes to ask whether (A G , A +,G ) → (A, A + ) is a finite morphism of complete Huber pair. Lemma 2.1.9 gives that the morphisms A G → A and A +,G → A + are integral. So we only need to show that A is finite as an The proof of Lemma 4.2.4 imitates the proof of the Adic Artin-Tate Lemma (Lemma 3.2.5), but it 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 3.2.5 before reading this proof. Proof.
Step 0. Preparation for the proof: We choose a pseudo-uniformizer ̟ ∈ R and an open, Step 1. We choose "good" A-module generators y 1 , . . . , y m of B: Remark B.2.10 implies that there is a compatible choice of rings of definition ̟ , we conclude that y 1 , . . . , y m are also A-module generators of B. The crucial property of this choice of A-module generators is that there exist a i,j , a i,j,k ∈ A 0 ⊂ A + such that Step 2. We define another ring of definition B 0 : We consider the unique surjective, continuous R-algebra homomorphism We define B 0 := g(R 0 X 1 , . . . , X n , Y 1 , . . . , Y m , T i,j , T i,j,k ), where R 0 is a ring of definition in R compatible with A 0 (see [Hub93b, Corollary 1.3(ii)] for its existence). This is clearly an open and bounded subring of B, so it is a ring of definition.
By construction, B 0 contains f (R 0 X 1 , . . . , X n ), and B 0 /̟B 0 is generated as an R 0 /̟R 0algebra by the classes x i , y j , a i,j , and a i,j,k .
Therefore, it is integral over R + B 0 since it contains R + f (R 0 X 1 , . . . , X n ) by the previous Step.
Step 4. We show that (B, B + ) is finite over (R T i,j , T i,j,k , R + T i,j , T i,j,k ): We recall that a i,j , a i,j,k ∈ A 0 ⊂ A + for all i, j, k. So, we can use the universal property of restricted power series to define a continuous morphism of complete Tate-Huber pairs: as the unique continuous R-algebra morphism such that We also define the morphism as the composition of r and i.
We now show that B 0 is finite over R 0 T i,j , T i,j,k . Note that this actually makes sense since the natural morphism factors through B 0 by the choice of B 0 . We consider the reduction B 0 /̟B 0 and claim that it is finite over Indeed, we know that B 0 /̟B 0 is generated as an R 0 /̟R 0 -algebra by the elements x 1 , . . . , x n , y 1 , . . . , 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 0 /̟B 0 is generated as an (R 0 /̟R 0 )[T i,j , T i,j,k ]-algebra by the elements x 1 , . . . , x n , y 1 , . . . , 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 coefficients in a i,j = t(T i,j ). This implies that B 0 /̟B 0 is generated as an (R 0 /̟R 0 )[T i,j , T i,j,k ]-algebra by y 1 , . . . , y m . But again, the same argument shows that each product y i y j can be expressed as a linear combination of y k with coefficients a i,j,k = t(T i,j,k ). This implies that y 1 , . . . , y m are actually (R 0 /̟R 0 )[T i,j , T i,j,k ]-module generators for B 0 /̟B 0 . Now we use a successive approximation argument (or [Sta21, Tag 031D]) to conclude that B 0 is finite over R 0 T i,j , T i,j,k . We conclude that B is a finite module over R T i,j , T i,j,k since Thus, we are only left to show that B + is integral over R + T i,j , T i,j,k .
Step 3 implies that B + is integral over B 0 R + , so it suffices to show that B 0 R + is integral over R + T i,j , T i,j,k . But this easily follows from the fact that B 0 is finite over R 0 T i,j , T i,j,k .
Step 5. We show that (A, A + ) is finite over (R T i,j , T i,j,k , R + T i,j , T i,j,k ): Note that R T i,j , T i,j,k is noetherian since R is strongly noetherian by assumption. Therefore, we see that A must be a finite R T i,j , T i,j,k -module as a submodule of a finite R T i,j , T i,j,k -module B. Moreover, we see that A + is equal to the intersection B + ∩ A because (B, B + ) is a finite (A, A + )-Tate-Huber pair. This implies that A + is integral over the image r(R + T i,j , T i,j,k ). We conclude that the complete Huber pair (A, A + ) is finite over (R T i,j , T i,j,k , R + T i,j , T i,j,k ). Therefore, it is topologically finite type over (R, R + ) by Lemma   Proof. Lemma 4.2.3 gives that A G , A +,G → (A, A + ) is a finite morphism of complete Tate-Huber pairs. So Lemma 4.2.4 guarantees that A G , A +G is a topologically finite type complete (R, R + )-Tate-Huber pair.
Theorem 4.2.6. Let (R, R + ) be a strongly noetherian complete Tate-Huber pair and X = Spa(A, A + ) a topologically finite type affinoid adic S = Spa(R, R + )-space with an S-action of a finite group G. Then the natural morphism φ : X/G → Y = Spa A G , A +G is an isomorphism over S. In particular, X/G is a topologically finite type affinoid adic S-space.
We adapt the proofs of Proposition 2.1.12 and 3.2.8. However, there are certain complications due 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 to work harder than in the 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 14 , and it is topologically finite type over S. Now we recall that there is a natural map of valuative 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 commutative diagram with the upper vertical maps being the forgetful maps and the lower vertical maps being the maps that 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 2.1.9 to find a prime ideal q ⊂ A that lifts the support Now we use [Mat86, Theorem 10.2] to dominate the valuation ring R v ⊂ k(p v ) by some valuation ring R w ⊂ k(q). This provides us with a valuation w : A → Γ w ∪{0} such that p ′ (w) = v. Therefore, the map p ′ is surjective.
14 The structure presheaf OY is a sheaf on Y by [Hub94, Theorem 2.5].
As for the transitivity of the G-action, we note that Lemma 2.1.9 implies that G acts transitively on the fiber (p ′′ ) −1 (p v ). Furthermore, [Bou98, Ch.5, §2, n.2, Theorem 2] guarantees that, for any prime ideal q ∈ (p ′′ ) −1 (p v ), the stabilizer subgroup surjects onto the automorphism group Aut(k(q)/k(p v )). We use [Bou98, Ch. 6, §8, n.6, Proposition 6] to see that there is a bijection between the sets 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 ideals of Nr k(q) (R v ). As a consequence, G q acts transitively on the set of valuations w ∈ p ′−1 (v) with the support q. Therefore, G acts transitively on p ′−1 (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 transitively on fibers: We recall that Spa(A, A + ) (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 ′−1 (v) is continuous and w(A + ) ≤ 1.
It is clear that w(A + ) ≤ 1 as A + is integral over A +,G . So we only need to show that the valuation w ∈ Spv(A) is continuous.
We choose a G-stable pair of definition (A 0 , ̟) from Lemma 4.2.1. Then [Bou98, Ch. 6, §8, n.1, Proposition 1] gives that Γ w /Γ v is a torsion group. Therefore w(̟) = v(̟) is cofinal in Γ w if it is cofinal in Γ v . In particular, w(̟) < 1. Now we verify the second condition in Lemma 4.2.7. Since w(A + ) ≤ 1 and v| A +,G = w| A +,G , Step 3. We show that φ : X/G → Y = Spa(A G , A +,G ) is a homeomorphism: Step 2 implies that φ is a bijection. Now note 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 morphism. The map π : X → X/G is a topological quotient morphism by its construction. Hence, φ is a homeomorphism.
Step 4. We show that φ is an isomorphism of topologically locally v-ringed spaces: Firstly, Remark B.1.2 implies that it suffices to show that the natural morphism is an isomorphism of sheaves of topological rings. Using the basis of rational subdomains in Y , it suffices to show that is a topological isomorphism for any f 1 , . . . , f n , s generating the unit ideal in A G . Lemma (7) is an (algebraic) isomorphism. We note that by Corollary B.3.7. Therefore, it suffices to show that  Our proof uses an adic analogue of the theory of "formal" models of rigid spaces in an essential way. It might be possible to justify this claim directly from first principles, but it seems quite difficult due to the fact that the complement X \ V does not have a natural structure of a pre-adic space.
In what follows, for any topological space Z with a map Z → Spec A + , we denote by Z the fiber product Z × Spec A + Spec A + /̟ in the category of topological spaces.
Proof. First of all, we note that rational subdomains form a basis of the topology of Spa(A, A + ), and they are quasi-compact. Therefore, we can find a quasi-compact open subspace F ⊂ V ′ ⊂ V , so we may and do assume that V is quasi-compact.
We consider the affine open immersion And define the category of U -admissible modifications Adm U,S to be the category of projective morphisms 16 f : Y → S that are isomorphisms over U . Then [Bha, Theorem 8.1.2 and Remark 8.1.8] (alternatively, one can adapt the proof of [FK18,Theorem A.4.7] to this situation) shows that  x. Then X/G is a locally topologically finite type adic S-space. Moreover, it satisfies the following properties: (1) π : X → X/G is universal in the category of G-invariant morphisms to topologically locally v-ringed S-spaces.
(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 morphism Z → X/G (see Definition B.4.1) of locally strongly noetherian analytic adic spaces, 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 Theorem 2.1.15, we can use Lemma 4.3.2 and Lemma 4.3.3 to reduce to the case of a strongly noetherian Tate affinoid S = Spa(R, R + ) and affinoid X = Spa(A, A + ). Then the claim follows from Theorem 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 is essentially trivial (Remark 2.1.4). 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 consider strongly noetherian Tate affinoid Z = Spa(B, B + ). Then the construction of the quotient implies that it suffices to show that the natural morphism of Tate-Huber pairs Remark 4.3.5. As the anonymous referee pointed out, the condition that each point x ∈ X admits an affinoid open neighborhood V x containing G.x may be automatic under a very mild assumption on X (it is probably enough to assume that X is separated over S). In the lemma below, we give a proof of this claim for separated rigid-analytic spaces. A more general version of this result would seem to require a generalization of the main results of [Tem00] to more general adic spaces. This is beyond the scope of this paper.
The next lemma uses Berkovich spaces in its proof; the reader unfamiliar with Berkovich spaces can safely ignore this lemma as it is never used in the rest of the paper. We heartfully thank Michael Temkin for suggesting the idea of the following argument: Lemma 4.3.6. Let K be a non-archimedean field with the residue field k, X a separated, locally finite type adic Spa(K, O K )-space, and {x 1 , . . . , x n } is a finite set of points of X. Then there is an open affinoid subset U ⊂ X containing all x i .
Proof. Firstly, we can replace X with a quasi-compact open subset containing all x i to reduce to the case of a quasi-compact (and separated) X. Then the underlying topological space of X is spectral, so [Sta21, Tag 0904] implies that it suffices to consider the case when all x i have the same (unique) rank-1 generalization x ∈ X.
Now since X is separated and quasi-compact, [ ω : X → X Ber that sends a point x to its unique rank-1 generalization. In what follows, we will slightly abuse the notation and consider x as both the point of X and X Ber (this is essentially harmless because X Ber is set-theoretically equal to the set of rank-1 points of X).
We consider the germ X Ber x (see [Ber93,§3.4]) and its reduction X Ber x = (V x , H(x), ε) (considered as an object of bir k , see [Tem00,p.4]) that is defined just after [Tem00, Lemma 2.1] (see also [Tem15, Definition 5.7.2.10]). Geometrically, the underlying topological space of the reduction X Ber x is identified with the fiber ω −1 (ω(x)) (see [Tem00, Remark 2.6]). In particular, points x i uniquely correspond to points of X Ber x that we also denote by x i (by slight abuse of notation  [Tem00,p.7] for the definition of a good germ and of a subdomain of a germ). Essentially by definition, this gives rise to a subdomain x ∈ W Ber ⊂ X Ber such that W is a good Berkovich space. Now the subdomain W Ber defines an open subspace W ⊂ X (by applying the functor (−) ad from [Hen16, Construction 7.5]) that contains all the points x 1 , . . . , x n by its construction. Therefore, we may replace X with W to assume that X Ber is good.

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 pseudouniformizer ̟.
We consider the functor of adic generic fiber: (−) k : admissible formal k + -schemes → Adic Spaces locally of topologically finite type over Spa(k, k + ) that is defined in [Hub96, §1.9] (it is denoted by d there). To an affine admissible formal k + -scheme Spf(A), this functor assigns the affinoid adic space Spa(A 1 ̟ , A + ) where A + is the integral closure of A in A 1 ̟ . Let X be an admissible formal k + -scheme with a k + -action of a finite group G. Then X k is a locally topologically finite type adic Spa(k, k + )-space with a Spa(k, k + )-action of G. Using the universal 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 natural morphism: Proof. Similarly to Step 1 in the proof of Theorem 3.4.1, the condition of Theorem 4.3.4 is satisfied for 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 Theorem 3.4.1 we can assume that X = Spf A is affine. Then we have to show that the natural map is an isomorphism of Tate-Huber pairs.
Clearly, A + is integral over A, and so it is integral over A G by Lemma 2.1.9(1). Hence,

Comparison of adic quotients and algebraic quotients.
For this section, we fix a complete, 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 over Spa(K, O K ). When we need to use classical rigid-analytic spaces, we refer to them as Tate rigidanalytic spaces.
In what follows, for any topologically finite type K-algebra A, we define Sp A := Spa(A, A • ). We note that [Hub94, Lemma 4.4] implies that for any affinoid space Spa(A, A + ) that is topologically finite type over Spa(K, O K ), we have A + = A • . So this notation does not cause any confusion.
We consider the analytification functor: (−) an : locally finite type K-schemes → Rigid Spaces over K that is defined as a composition (−) an = r K • (−) rig of the classical analytification functor (−) rig (as it is defined in [Bos14, §5.4] and the functor r K that sends a Tate rigid space to the associated adic space (see [Hub94,§4]). The main issue with the analytification functor is that it does not send affine schemes to affinoid spaces. More precisely, the analytification of an affine scheme X = Spec K [T 1 , . . . , T d ] /I is canonically isomorphic to 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 can assume that X = Sp A is 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
Finally, we note that the natural map is an isomorphism.
This implies that the m x -adic topology on 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 The claim now follows from Lemma 2.1.11 and Lemma 4.5.1. 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 2.1.13 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 by G-stable adic subspaces. Now we choose i such that x ∈ U an i . We note that (9) implies that each U an i can be written as an ascending union of open affinoid subspaces. Since the orbit G.x is finite, it is contained in some U (n) i .
Step 2. φ is a bijection on classical points: Classical points of (X/G) an are identified with (X/G) 0 the set of closed points of X/G (see [Con99,Lemma 5.1.2(1)]). Thus the noetherian analogue of Theorem 3.3.4 (or Theorem A.2.9) and the fact that finite morphisms reflect closed points imply that classical points of (X/G) an are identified with X 0 /G.
Similarly, we can use Theorem 4.3.4 and the fact that π X an reflects classical points (as being finite) to conclude that the classical points of X an /G are identified with the set X 0 /G.
Step 3. We reduce to a claim on completed local rings: We consider the commutative diagram: X an Since π an X is a finite, surjective, G-equivariant morphism, we conclude that φ is a finite, surjective morphism by Proposition 5.3.1 (that is independent of the present section). Therefore, φ * (O X an/G ) is a coherent O (X/G) an -module by Corollary B.5.3 (2) and it suffices to show is an isomorphism. Since classical points on a rigid-analytic space reflect isomorphisms of coherent sheaves 19 , it suffices to show that O (X/G) an ,y → φ * O X an /G y is an isomorphism for any classical point y ∈ (X/G) an . Now we use [Con06, Theorem A.1.3], finiteness and surjectivity of φ (that, in particular, implies that φ is surjective on classical points), and Step 2 to conclude that it suffices to show that the natural map is an isomorphism at each classical point of X an /G. We note that φ # x is a local homomorphism of noetherian local rings by [Bos14,Proposition 4 x is an isomorphism, where the completions on both sides are understood with respect to the corresponding maximal ideals.
Step 4. We show that φ # x is an isomorphism: We note Corollary 4.5.2 gives that Now we consider the natural morphism of locally ringed spaces i : (X /G) an → X /G. By [Con99, Lemma 5.1.2(1), (2)], i is a bijection between the sets of classical points of (X/G) an and closed points of X/G, and the natural morphism is an isomorphism for any closed point y ∈ (X/G) an .

Properties of the Geometric Quotients
We discuss some properties of schemes (resp. formal schemes, resp. adic spaces) that are preserved 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 explicitly constructed only in the affine case, and in general one needs to do some gluing to get X/G. This gluing might, a priori, destroy certain global properties of X such as separatedness. In this section we show that this does not happen for many geometric properties in all schematic, formal and adic setups. We mostly stick to the properties we will need in our paper [Zav21a].

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 to other settings (formal schemes and adic spaces).
Let f : X → Y be a G-invariant 21 morphism of flat, locally finite type k + -schemes. We assume that any orbit G.x ⊂ X lies inside some open affine subscheme V x ⊂ X. In particular, the conditions of Theorem 2.1.15 are satisfied, so it gives that X/G is a k + -scheme with a finite 20 One can repeat the proof of Corollary 4.5.2 using that Lemma 4.5.1 holds on the level of henselian local rings in the scheme world. 21 Here and later in the paper, a G-invariant morphism means a G-equivariant morphism for some G-action on the source and a trivial G-action on the target. morphism π : X → X/G. The universal property of the geometric quotient implies that f factors through π and defines the commutative diagram: 1. Let f : X → Y , a finite group G, and f ′ : X/G → Y be as above. Then f ′ is quasi-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 commutes with 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 see that 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 2.1.15, we conclude that X/G is quasi-compact. Therefore, f ′ is quasi-compact as well.
Quasi-separatedness: We suppose that the diagonal morphism ∆ X : X → X × Y X is quasicompact and consider the following commutative square: We know that π is finite, so it is quasi-compact. Therefore, the morphism π× Y π is quasi-compact as 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 Diagram (11) once again. Since π is finite we conclude that π × Y π is finite as well, hence closed. Now we use surjectivity of π to get an equality: with (π× Y π)(∆ X (X)) being closed as image of the closed subset ∆ X (X) (by separateness of X over affine Y ). This shows that ∆ X/G (X/G) is a closed subset of X/G × Y X/G, so X/G is separated.
Properness: We already know that properness of f implies that f ′ is quasi-compact and separated. Also, Theorem 2.2.6 shows that f ′ is locally of finite type, so it is of finite type. The only thing that we are left to show is that it is universally closed. But this easily follows from universal closedness of f and surjectivity of π.
Finiteness: A finite morphism is proper, so the case of proper morphisms implies that f ′ is proper. It is also quasi-finite as π is surjective and f = f ′ • π has finite fibers. Now Zariski's Main Theorem [EGA IV 4 , Corollaire 18.12.4] implies that f ′ is finite.
We now slightly generalize Proposition 5.1.1 to the case of a G-equivariant morphism f . Namely, we consider a G-equivariant morphism f : X → Y of flat, locally finite type k + -schemes. We assume that the actions of G on both X and Y satisfy the condition of Theorem 2.2.6. Then the universal property 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. a k-modification 22 , resp. finite) if f is so.
Proof. We start the proof by considering the commutative diagram Note that, for all relevant properties P except for k-modification, f satisfies P implies that h satisfies P due to finiteness of π Y . Thus, 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 from the fact that the formation of the geometric quotient commutes with flat base change such as Spec k → Spec k + .
Lemma 5.1.3. Let Y be a flat, locally finite type k + -scheme, and f : X → Y a G-torsor for a finite group G. The natural morphism f ′ : X/G → Y is an isomorphism.
Proof. Since a G-torsor is a finiteétale 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 the action on Y is trivial. Thus, X/G is a flat, locally finite type k + -scheme. The universal property of the geometric quotient defines the map X/G → Y that we need to show to be an isomorphism. It suffices to check thisétale locally on Y as the formation of X/G commutes with flat base change by Theorem 2.1.15 (4). Therefore, it suffices to show 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 obvious and follows either from the construction or from the universal property.

Properties of the Formal Quotients.
Similarly to Section 5.1, we discuss that certain properties descend to the geometric quotient in the formal setup. Most proofs are similar to those in Section 5.1.
For the rest of the section, we fix a complete, microbial valuation ring k + with a pseudouniformizer ̟ and field of fractions k.
We consider a G-equivariant morphism f : X → Y of admissible formal k + -schemes. We assume that the actions of G on both X and Y satisfy the condition of Theorem 3.3.4. Then the universal property of the geometric quotient implies that f descends to a morphism f ′ : 22 A morphism f : X → Y of flat, locally finite type k + -schemes is called a k-modification, if it is proper and the base change f k : X k → Y k is an isomorphism.
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. a rig-isomorphism 23 , 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 in place of Theorem 2.2.6. The rig-isomorphism case is easy, akin to the k-modification case in Proposition 5.1.2. We only need to use Theorem 4.4.1 in place of Theorem 2.1.15 (4).
Now suppose that f is finite. The proper case implies that f ′ is proper, and it is clearly quasifinite. Therefore, the mod-̟ fiber 3] gives that f ′ is finite.
Lemma 5.2.2. Let Y be an admissible formal k + -scheme, and f : X → Y a G-torsor for 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 used is that one can check that a morphism is an isomorphism after a finiteétale base change (and we use Theorem 3.3.4(4) in place of Theorem 2. 1.15(4)). This follows from descent for adic, faithfully flat morphisms [FK18, Proposition I.6.1.5] 24 5.3. Properties of the Adic Quotients. Similarly to Section 5.1 and Section 5.2, we discuss that certain properties descend to the geometric quotient in the adic setup.
For the rest of the section, we fix a locally strongly noetherian analytic adic space S. We consider a G-equivariant S-morphism f : X → Y of locally topologically finite type adic Sspaces. We assume that the actions of G on both X and Y satisfy the condition of Theorem 4.3.4. Then the universal property of the geometric quotient implies that f descends to a morphism f ′ : X/G → Y /G over S: 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. Lemma 5.3.2. Let Y be a locally topologically finite type adic S-space, and f : X → Y a G-torsor for a finite group G. The natural morphism f ′ : X/G → Y is an isomorphism.
23 A morphism f : X → Y of admissible formal k + -schemes is called a rig-isomorphism if the adic generic fiber The case of a finite flat morphism is much easier as the completed tensor product along a finite module coincides with the usual tensor product due to Lemma 3.1.6(1).
Proof. The proof of Lemma 5.1.3 adapts to this situation. The only non-trivial fact that we used is that one can check that X/G → Y is an isomorphism after a surjective, flat base change (and we use Theorem 4.3.4(4) in place of Theorem 2. 1.15(4)). This follows from Lemma B.4.10 as f ′ is finite due to Proposition 5.3.1.
A.2. Universally Adhesive Schemes. Remark A.2.2. A valuation ring k + is universally adhesive if it is microbial (in the sense of Definition 3.1.1). More precisely, k + is universally ̟-adically adhesive for any choice of a pseudouniformizer ̟ ∈ k + . Indeed, [FK18, Proposition 0.8.5.3] implies that it is sufficient to see that any finite k + [X 1 , . . . , 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 over k + 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 if and only if k + is microbial.
Proof. If k + is microbial, we take I = (̟) for any pseudo-uniformizer ̟. Then k + is I-adically adhesive by Remark A.2.2. Now we suppose that k + is adhesive for some finitely generated ideal I. Then I = (a) is principal because k + is a valuation ring. Hence, k + 1 a is a noetherian valuation ring by the I-adic adhesiveness of k + . Therefore, k + 1 a is either a field or discrete valuation ring. We firstly consider the case k + 1 a is a field. We then observe that rad(a) is a height-1 prime ideal of k + by [FK18, Proposition 0.6.7.2 and Proposition 0.6.7.3]. Therefore, k + is microbial by [Hub96, Definition 1.1.4] or [Sem15, Proposition 9.1.3].
Now we consider the case k + 1 a 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.
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 M a finite A-module.
(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 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 1 , . . . , X d ] → A. Then the definition of universal adhesiveness says that R[X 1 , . . . , X d ] is I-adically adhesive. This easily implies that so is A. Now the first two claims follow from [FK18, Proposition 0.8.5.3]. To show the last claim, we note that the kernel ϕ is a saturated submodule of R[X 1 , . . . , X d ], so it is a finitely generated ideal by Part (1). Therefore, A is finitely presented as an R-algebra.
Lemma A.2.5. Let (R, I) be a universally adhesive pair (see Definition A.2.1), and A → B be a finite 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 proof is analogous to that of Lemma 2.2.3 and again the only difficulty lies in showing that A is finite over A ′ (as defined in the proof of Lemma 2.2.3). This follows from Lemma A.2.4(1).
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 the natural morphism A G → A is finitely presented. 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. Suppose that each point x ∈ X admits an affine neighborhood V x containing G.x. Then the scheme X/G as in Theorem 2.1.15 is I-torsion free and locally finite type over S, and the integral surjection π : X → X/G is finite and finitely presented. An adically topologized ring R endowed with the adic topology defined by a finitely generated ideal of definition I ⊂ R is topologically universally adhesive if R is I-adically complete, and the pair (R, I) is topologically universally adhesive.
Remark A.3.2. We note that the definition of topologically universally adhesive topological rings is independent of the choice of a finitely generated ideal of definition. For any two ideals 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 by Remark 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 pseudouniformizer ̟ ∈ k + . This is proven in [FK18, Theorem 0.9.2.1]. Alternatively, one can easily show the claim from Lemma 3.1.6 and the classical fact that k is strongly noetherian, i.e. k X 1 , . . . , X d is noetherian for any d ≥ 0.
Lemma A.3.4. Let R be a complete, topologically universally I-adically adhesive ring, A be a topologically 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 topology on N .
(3) Let N ⊂ M be a saturated A-submodule of M . Then N is a finite A-module.
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 generated ideal of definition I ⊂ A.
We show that this definition is independent of the 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 of the choice of a finitely generated ideal of definition I. Thus, using that X is quasi-compact, we can assume that Lemma A.3.14. Let S be a universally adhesive formal scheme, and let X be an S-finite, admissible formal S-scheme. Suppose that S ′ → S is a flat adic morphism of universally adhesive formal schemes. 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 assume that S, S ′ (and, therefore, X) are affine. Suppose S = Spf A, S ′ = Spf A ′ , and X = Spf B for an A-algebra B finite as an A-module (see [FK18, Proposition I.4.2.1]). Choose an ideal of definition I ⊂ A, our assumptions imply that IA ′ is an ideal of definition in A ′ . We know that X ′ is given by Spf A ′ ⊗ A B, so it is easily seen to be a topologically finite type formal S ′ -scheme. We are only left to check that A ′ ⊗ A B is I-torsionfree.
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 Remark A.3.11 ensures that A ′ is A-flat, and so A ′ ⊗ A B is B-flat. Since B had no I-torsion, the same holds for A ′ ⊗ A B.
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 finite group 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 locally ringed S-spaces.
(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 universally adhesive formal scheme Z and a flat adic morphism Z → X/G, the geometric quotient (X × X/G Z)/G is a formal schemes, and the natural morphism (X × X/G Z)/G → Z is an isomorphism.
Proof. The proofs of parts (1), (2) We explain part (4) in a bit more detail. We first 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. Then Lemma A.3.14 implies that X ′ is S ′ -admissible, and then one can repeat the proof of Theorem 3.3.4 using Lemma A.3.4(1) in place of Lemma 3.1.6(1).
Theorem A.3.16. Let R be an topologically universally I-adically adhesive ring, and X an Itorsion free, locally finite type R-scheme with a R-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 I-adic completion X with the induced R + -action of G, and the natural morphism Proof. The proof of Theorem 3.4.1 goes through in this wider generality. The only new non-trivial input is flatness of A G → 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 (see Definition [FK18, Section 0.8.2(a)]) by [FK18,Proposition 0.8.5.16]. Therefore, [FK18,8.2.18(i)] implies that B → B is flat.

Appendix B. Foundations of Adic Spaces
The theory of adic spaces still seems to lack a "universal reference" for proofs of all basic questions one might want to use. For example, all of [Hub93b], [Hub94], [Hub96] and [KL19] do not really discuss notions of flat and separated morphisms in detail. The two main goals of this Appendix are to provide the reader with the main definitions we use in the paper, and to give proofs of claims that 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 26 since this is the only case that we need in this paper.
B.1. Basic Definitions. We start this section by recalling the definition of the category of topologically locally v-ringed spaces V. This category will play the same role as the category of locally ringed spaces plays for the category of schemes. Namely, V is going to be a sufficiently flexible category with a fully faithful embedding of the category of adic spaces into it.
Definition B.1.1. A category of topologically locally v-ringed spaces 27 V is the category 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 ringed spaces 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).
Remark B.1.2. The category V comes with the forgetful functor F : V → TLRS to the category of topologically locally ringed spaces. It is clear that this functor is conservative.
Definition B.1.3. We define the category AS of analytic adic spaces as the full subcategory of V whose objects are triples (X, O X , {v x } x∈X ) locally isomorphic to Spa(A, A + ) for a complete Tate-Huber pair (A, A + ). We remind the reader that this requires the pair (A, A + ) to be "sheafy". In the following, if we consider a Tate affinoid adic space X = Spa(A, A + ), we implicitly assume that (A, A + ) is a complete Tate-Huber pair. There are crucial properties of topologically finite type morphisms that makes them behave similarly to finite type morphisms: Proof. This is [Hub94,Lemma 3  We claim that the natural morphism A 0 → B 0 is finite. It suffices to prove that it is finite mod ̟ by successive approximation and completeness. However, it is clearly finite type mod ̟ since it coincides with the composition: and it is integral since B 0 /̟B 0 is algebraically generated over A 0 /̟A 0 by the residue classes b 1 , . . . , b n that are integral over A 0 /̟A 0 by construction. Thus this map is integral and finite type, hence finite.
Finally, (A, Remark   An analytic adic space S is strongly noetherian if it is locally strongly noetherian and quasicompact. Proof. The "if" direction is clear. Now suppose that X := Spa(A, A + ) is strongly noetherian as an analytic adic space. We wish to show that the Tate-Huber pair (A, A + ) is strongly noetherian.
By assumption, each point x ∈ X has an affinoid open neighborhood U x = Spa(A x , A + x ) with a strongly complete Tate-Huber pair (A x , A + x ). Since rational subdomains form a basis of topology of X and strong noetherianess is preserved by passing to rational subdomains (see Lemma B.2.14), we can assume that each U x ⊂ X is a rational subdomain. B.3. Completed Tensor Products. The main goal of this section is to prove that under certain assumptions, completed tensor products of Tate rings coincide with usual tensor products. This should be well-known to the experts, but it seems difficult to extract the proof from the existing literature.
For the rest of the section, we fix a complete Tate-Huber pair (A, A + ) with a choice of a pair of definition (A 0 , ̟). We recall the notion of "the natural A-module topology" for a finite A-module M : 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 prove the claim in the case n = 1 by the universal property of direct products. But any A-linear map A → N to a topological A-module is clearly continuous.
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-module structure. We want to show that any A-linear morphisms g : M → N to a topological A-module N is continuous. We consider the diagram: As for the second claim, we use Lemma B.2.9 to find rings of definition A 0 , B 0 such that B 0 is finite over A 0 . Choose some generators b 1 , . . . , b n for B 0 over A 0 and consider the morphism p : A n → B that sends (a 1 , . . . , a n ) to a 1 b 1 + · · · + a n b n . Then clearly p(A n 0 ) = B 0 . Finally, we recall that given two morphisms f : A → B and g : A → C of Tate rings there is a canonical way to topologize the tensor product B ⊗ A C. Namely, we pick some rings of definition B 0 ⊂ B and C 0 ⊂ C such that f (A 0 ) ⊂ B 0 and g(A 0 ) ⊂ C 0 . Then we topologize A ⊗ B C by requiring the image (B ⊗ A C) 0 := Im(B 0 ⊗ A 0 C 0 → B ⊗ A C) with its ̟-adic topology to be a ring of definition in B ⊗ A C. Then it is straightforward 29 to see that the Tate ring B ⊗ A C ring satisfies the expected universal property in the category of Tate rings. In particular, this shows that this construction does not depend on the choice of rings of definition A 0 , B 0 , C 0 . 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 denoted by B ⊗ A C. 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 0 ⊂ B and a surjection p : A n → B such that p(A n 0 ) = B 0 . Then after tensoring it with C we get a surjective morphism C n → B ⊗ A C, and tensoring the surjection A n 0 → B 0 with C 0 we get a surjection C n 0 → B 0 ⊗ A 0 C 0 . Combining these, we get a commutative diagram: 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 some facts that we are using in the paper. Similarly to the case of formal schemes, we will soon describe flatness of strongly noetherian Tate 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 the topology on an affinoid space, so O X,x is equal to the 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 preserved by filtered colimits, we conclude that A → O X,x is flat. Note that this implies that A p → O X,x is flat as well. Indeed, this easily follows from the fact that for any A p -module M we have isomorphisms The discussion above shows that r x : A p → O X,x is flat, but we also need to show that it is faithfully flat. In order to prove this claim it suffices to show that A p → O X,x is a local ring homomorphism. We recall that the maximal ideal m x ⊂ O X,x is given as We need to show r x (pA p ) ⊂ m x . We pick any element h ∈ pA p . It can be written as f /s for f ∈ p and s ∈ A \ p, and we need to check that v f s = 0. The very definition of p as the support of v implies that v (f ) = 0 and v (s) = 0. Thus where w = f (x) ∈ Spa(A, A + ). We use Lemma B.4.2 to conclude that we have a commutative square 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 f ♯ : A → B is flat. Step 2. Finish the argument: We pick a point x ∈ X with the image y = f (x). Then the morphism f ♯ y : O Y,y → O X,x can be rewritten as . Therefore, it suffices to show that the natural morphism Remark B.4.8. One can also show that X L → X is flat for any extension of non-archimedean fields K ⊂ L and a rigid-analytic K-space X. Again, the main input is to show that the moprhism (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 noetherian Remark B.6.4. If i : X → Y is a closed immersion of (locally strongly noetherian) adic spaces with I = ker(O Y → i * O X ), then there is a set-theoretic identification: Lemma B.6.5. Let Y = Spa(A, A + ) be a strongly noetherian Tate affinoid 30 , and i : X → Y a closed immersion. Then B := O X (X) is a complete Tate ring, and the natural morphism i * : A → B is a topological quotient morphism.
Proof. The natural morphism A → B is clearly continuous as O Y → i * O X is a morphism of sheaves of topological rings. Therefore, any topologically nilpotent unit ̟ A ∈ A defines a topologically nilpotent unit ̟ := i * (̟ A ) ∈ B. We now show that B is a Tate ring. Since X is closed in an affinoid, we conclude that X is quasi-compact and quasi-separated. So we choose a finite covering X = ∪ n i=1 U i by open affinoid U i = Spa(B i , B + i ). Then Recall that we always implicitly assume that (A, A + ) is complete. and the topology on B coincides with the subspace topology. Each B i admits a ring of definition B i,0 , and we can assume that the topology on every B i,0 is the ̟-adic topology (possibly after replacing ̟ A with a power). We claim that is a ring of definition in B. It suffices to show the topology on B 0 induced from n i=1 B i,0 coincides with the ̟-adic topology. This follows from the equalities that, in turn, follow from the fact that ̟ is invertible in B.
Now we address completeness of B. By a similar reason as above, we see that there is a short exact sequence such that d is a topological embedding and a is continuous. Using that X is quasi-separated, we can cover each U i ∩ U j by a finite number of affinoids V i,j,k . Thus, we get a short exact sequence 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 the complete Tate ring n i=1 B i . Thus, it is also complete. Finally, we show that A → B is a topological quotient morphism. We consider the short exact sequence  Lemma B.6.6 implies that (A/I, (A + /I ∩A + ) c ) is a complete Tate-Huber pair. Thus Lemma B.3.5 ensures that the tensor product topology on (A/I) ⊗ A B coincides with the natural topology. Lemma B.6.6 also implies that B/IB is a complete Tate ring, in particular, its topology is first countable. Thus, Lemma B.3.4 ensures that its topology coincides with the natural topology. Therefore, the universal property of the natural topology guarantees that the canonical algebraic isomorphism (A/I) ⊗ A B ≃ B/IB preserves topologies on both sides. Now we recall that Corollary B.6.9. Let Y = Spa(A, A + ) be a strongly noetherian Tate affinoid, and let i : X → Y be a 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 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 is an isomorphism of sheaves of topological rings. Corollary B.6.8 implies that this is an algebraic isomorphism. We use Remark B.2.3 and Lemma B.6.5 32 to handle the topological aspect of the isomorphism.
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 → Z is a closed immersion, . 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.
Remark B.6.13. The order of the open and closed immersion in Definition B.6.11 is needed to ensure 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 that the image f (X) is closed in Y . Then f is a closed immersion.
Proof. We write f as a composition We use the open covering Y = U ∪ (Y \ f (X)). We know that I| U is coherent by assumption, and it is clear that I| Y \f (X) ≃ 0 is coherent. Therefore, we conclude that I is coherent on Y . Now we show surjectivity of f ♯ . We note that since f is topologically a closed embedding, we conclude 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.
Definition B.7.1. We say that a locally topologically finite type morphism f : X → Y of locally strongly noetherian analytic adic spaces is separated, if the diagonal morphism ∆ X/Y : X → X × Y X has closed image Remark B.7.2. We assume that f is locally topologically finite type to ensure the existence of the fiber product X × Y X.
Lemma B.7.3. Let f : X → Y be a locally topologically finite morphism of locally strongly noetherian 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 −1 (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 that contains ∆ X/Y (X). Thus in order to show that ∆ X/Y is an immersion, it is suffices to show is a closed immersion. Moreover, we note that α −1 (V i,j × U i V i,j ) = V i,j for any i ∈ I, j ∈ J i . Since the notion of a closed immersion is easily seen to be local on the target, we conclude that it is enough to show that the diagonal 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 B ⊗ A B, B ⊗ A B +