Reified valuations and adic spectra

We revisit Huber's theory of continuous valuations, which give rise to the adic spectra used in his theory of adic spaces. We instead consider valuations which have been reified, i.e., whose value groups have been forced to contain the real numbers. This yields reified adic spectra which provide a framework for an analogue of Huber's theory compatible with Berkovich's construction of nonarchimedean analytic spaces. As an example, we extend the theory of perfectoid spaces to this setting.

There are several frameworks for analytic geometry over nonarchimedean fields, which can be classified into roughly three types: • rigid analytic geometry (Tate), which can also be obtained via formal geometry with admissible blowups (Raynaud); • nonarchimedean analytic geometry (Berkovich), which can also be obtained via tropical geometry (Payne et al.); • adic geometry (Huber), which can also be obtained via formal geometry (Abbes, Fujiwara-Kato). For a comparative discussion (primarily between the first two viewpoints), see [6]. Here, we limit ourselves to an instructive analogy: the three frameworks give results analogous to those of the following three constructions.
• Consider the rational numbers with the Grothendieck topology of finite unions of closed intervals with rational endpoints. Let T 1 be the resulting topos. • Consider the real numbers. The Grothendieck topology of finite unions of closed intervals with rational endpoints recovers T 1 . The natural topology defines a new topos T 2 . The Grothendieck topology of finite unions of all closed intervals defines a new topos T 3 . • Consider the real numbers plus some additional points r ± ǫ for each rational number r. The natural topology recovers the topos T 1 . In this paper, we introduce a construction playing the role of the real numbers plus points r±ǫ for each real number r, whose natural topology recovers the topos T 3 . This makes it possible to overcome a mismatch between the theories of Berkovich and Huber: while Huber's theory is based on the classical theory of Krull valuations, Berkovich's theory is based on real-valued seminorms. The link comes via the fact that any rank 1 Krull valuation can be interpreted as a real valuation; however, one can rescale a real valuation without changing the equivalence class of the underlying Krull valuation. To correct this, we consider reified valuations, for which we fix the comparisons between real numbers and elements of the value group. These also appear in upcoming work of Ducros and Thuillier on the relationship between monomial valuations and skeleta of Berkovich spaces [7].
Using reified valuations, one can simulate much of the analysis of continuous valuations from [16] and the comparison of rigid and adic spaces by Huber [16, §4] and van der Put and Schneider [27]. In fact, in some ways the reified version of the analysis is somewhat simpler. For example, when working with Banach algebras over an ultrametric field, the case of a trivially valued field can be handled more uniformly using reified valuations; this is consistent with the corresponding uniformity in Berkovich's theory. Roughly speaking, reification provides an alternative to the use of topologically nilpotent units, such as in Tate's fundamental theorem on the acyclicity of the structure sheaf.
We also describe the structure presheaf on a reified adic spectrum and carry out some of the local preliminary work to a theory of reified adic spaces. As in Huber's construction of adic spaces, the construction of reified adic spaces involves topological rings plus some auxiliary data. In Huber's construction, the auxiliary datum associated to a topological ring is a certain subring of integral elements; the analogous datum in our setting is defined in terms of the graded ring associated to a nonarchimedean Banach space. (The graded ring first appeared prominently in work of Temkin extending some key properties of rigid analytic spaces to Berkovich spaces [26], so its appearance here is perhaps not surprising.) For the reified adic space associated to a single ring, we establish a Tatestyle acyclicity theorem for the structure sheaf and a Kiehl-style glueing theorem for vector bundles (and for coherent sheaves under a suitable noetherian hypothesis), following [20, §2]. One important point is that in the context of Berkovich spaces, these results apply to coverings for the full Gtopology, as shown in [2]; by contrast, by passing from Berkovich spaces to adic spaces, one only obtains acyclicity and glueing with respects to coverings for the strictly analytic G-topology. For an explicit example, pick 0 < ρ 2 ≤ ρ 1 and consider the disc |T | ≤ ρ 1 : in the full G-topology this disc admits an admissible covering by the disc |T | ≤ ρ 2 and the annulus ρ 2 ≤ |T | ≤ ρ 1 ; in the strictly analytic G-topology, this only occurs if ρ 1 belongs to the divisible closure of the norm group of the base field.
As an illustration, we describe the spaces associated to perfectoid algebras; this amounts to a fairly faithful translation of certain sections of [20]. In fact, this paper was borne out of the author's frustration with the status quo during the writing of [20]: while in many respects it is natural to study perfectoid algebras via their Gel'fand transforms, these cannot be easily glued without promoting them to something like adic spaces, and at the time no such construction was available in the literature.
To conclude this introduction, we mention two related constructions. Instead of fixing comparisons between elements of the value group and arbitrary positive real numbers, one may only fix these comparisons for real numbers in some multiplicative subgroup H; this yields the concept of Hreified valuations, which interpolates between ordinary valuations and our reified valuations. One can easily modify our arguments to produce statements about such valuations, but we have not done so (despite such valuations making an appearance in [7]). In a different direction, Foster and Ranganathan [8] have described an analogue of tropicalization in which the real numbers are replaced by the value group of a valuation of possibly higher rank; reified valuations provide a natural context for comparing this construction to ordinary tropicalization.

Spectral and prespectral spaces
We first generalize Hochster's formalism of spectral spaces [14] to Gtopological spaces. This links the spaces considered by Huber with other types of analytic spaces; see for example [16, §4]. Definition 1.1. A bounded distributive lattice is a partially ordered set D satisfying the following conditions.
(a) The set D has a least element 0 and a greatest element 1.
(b) For any x, y ∈ D, the set of z ∈ D for which z ≤ x, z ≤ y has a greatest element x ∧ y (the meet of x and y). (c) For any x, y ∈ D, the set of z ∈ D for which z ≥ x, z ≥ y has a unique least element x ∨ y (the join of x and y). (d) The meet and join operations are distributive over each other. Let DLat denote the category whose objects are bounded distributive lattices and whose morphisms are maps of sets preserving ≤, 0, 1, ∧, ∨. Definition 1.2. A filter on D ∈ DLat is a subset F of D satisfying the following conditions.
(a) We have 1 ∈ F and 0 / ∈ F. (b) For any S 1 , S 2 ∈ F, we have S 1 ∧ S 2 ∈ F. (c) For any S ∈ F, any T ∈ D with T ≥ S is also in F. A filter F on D is prime if it satisfies the following additional condition.
Let Spec(D) be the set of prime filters on D equipped with the topology generated by the setsS := {F ∈ Spec(D) : S ∈ F} for S ∈ D.
We now recall some relevant properties of G-topological spaces. Definition 1. 3. Let X be a G-topological space in the sense of [3, Definition 9.1.1/1], i.e., a Grothendieck topology whose underlying category is a family of subsets of X closed under pairwise intersections. (In practice, it is harmless to assume that the family is closed under finite intersections, as this only adds the condition that X itself is an open subset.) We say X is T 0 if for any x = y ∈ X, there exists an open subset of X containing exactly one of x, y.
A nonempty closed subspace Z of X is irreducible if for any two open subsets U, V such that U ∩ Z and V ∩ Z are nonempty, U ∩ V ∩ Z is also nonempty; it is enough to check this for U, V running through a basis.
The closure Z of a point x ∈ X is irreducible: an open set meets Z if and only if it contains x. We say X is sober if conversely any irreducible closed subset of X is the closure of a unique point of X; any sober space is T 0 .
We now introduce the concepts of spectral and prespectral spaces. We say that X is quasiseparated (or semispectral in the language of [14]) if the intersection of any two quasicompact open subsets of X is again quasicompact. We write qcqs as shorthand for quasicompact and quasiseparated.
We say that X is prespectral if X is qcqs, any finite union of quasicompact open sets is open, and the quasicompact open sets form a basis. In particular, the quasicompact open subsets of X form a bounded distributive lattice with 0 = ∅, 1 = X, ∧ = ∩, ∨ = ∪.
We say that X is spectral if the G-topology on X is an ordinary topology (that is, any union of open sets is open) and X is both prespectral and sober. Spectral spaces are called coherent spaces in some sources, such as [21].
A map f : X → Y between prespectral spaces is spectral if the preimage of any quasicompact open subset is a quasicompact open. If X is a topological space, this forces f to be continuous. Let Prespec (resp. Spec) be the category of prespectral (resp. spectral) spaces and spectral morphisms.
The key property of spectral spaces is the following result of topos theory. Proof. The functor DLat op → Spec acts on morphisms via pullback: for f : D 1 → D 2 a morphism in DLat and F ∈ Spec(D 2 ), the set {S ∈ D 1 : f (S) ∈ D 2 } is a prime filter on D 1 . For more, see [21,Corollary II.3.4].
Corollary 1.6. The forgetful functor Spec → Prespec admits a left adjoint taking X ∈ Prespec to Spec(D(X)) for D(X) the lattice of quasicompact open subsets of X.
Proof. For X ∈ Prespec, the adjunction map X → Spec(D(X)) takes x ∈ X to the prime filter {S ∈ D(X) : x ∈ S}. (Note that this map is spectral, but not necessarily continuous if X is not an ordinary topological space.) On the other side, for Y ∈ Spec, Theorem 1.5 provides a natural isomorphism Spec(D(Y )) ∼ = Y for which the composition Y → Spec(D(Y )) → Y is the identity map. Definition 1.7. For X a topological space, the patch topology (or constructible topology) on X is the new topology generated by the open sets and complements of quasicompact open sets of the original topology. If X is a spectral space, we sometimes call its original topology the spectral topology to distinguish it from the patch topology.
The key property of the patch topology is the following [14, Theorem 1]. Theorem 1.8. Any spectral space is compact under the patch topology.
Proof. It is clear that the patch topology is Hausdorff. To check quasicompactness, it suffices to check that any family of closed and quasicompact open sets for the spectral topology which is maximal for the finite intersection property has nonempty intersection. But the intersection of the closed members of such a family is irreducible (by maximality) and so has a generic point, which belongs to the full intersection.  Corollary 1.11. A topological space which is T 0 and prespectral is spectral if and only if its patch topology is quasicompact.
Proof. For any irreducible closed subspace Z, any point in the intersection of the quasicompact open subsets of Z is a generic point. Remark 1.12. We collect some additional observations about the adjunction map X → Spec(D(X)) associated to X ∈ Prespec via Corollary 1.6.
(a) This map is the unique (up to unique isomorphism) morphism f :

Spaces of valuations
Throughout §2, fix a (commutative unital) ring A. We recall the construction and basic properties of the space of valuations on A, following [16]. Given our goals, it is natural to write valuations and semivaluations multiplicatively rather than additively; this is inconsistent with classical literature on valuation theory, but it is consistent with Huber's papers.
Definition 2.1. By a value group, we will mean a totally ordered abelian group written multiplicatively (so that 1 is its identity element). For Γ a value group, let Γ 0 denote the pointed commutative monoid Γ ∪ {0} ordered so that 0 < γ for all γ ∈ Γ. Definition 2.2. A semivaluation on the ring A is a function v : A → Γ 0 for some value group Γ satisfying the following conditions.
For v a semivaluation, let Γ v,0 be the image of v and put Γ v := Γ v,0 \ {0}. Two semivaluations v 1 , v 2 on A are equivalent if there exists an isomorphism i : Γ v 1 ∼ = Γ v 2 of value groups (which we also view as an isomorphism i : The equivalence classes of semivaluations on A then correspond to pairs (p, o) in which p is a prime ideal of A and o is a valuation ring of Frac(A/p). Definition 2.3. The valuative spectrum of A is the set Spv(A) of equivalence classes of semivaluations on A, equipped with the topology generated by sets of the form Let B be the Boolean algebra generated by sets of the form (2.4); note that B is also generated by the sets it is injective with image defined by closed conditions (see [ is surjective (but typically not injective; see Remark 2.10).
Proof. To prove (a), see [28, §5]. Fix v 1 ∈ Spv(k 1 ), v 2 ∈ Spv(k 2 ) which both restrict to v ∈ Spv(k); we must exhibit a common overfield k 3 of k 1 and k 2 and an element v 3 ∈ Spv(k 3 ) mapping to v 1 ∈ Spv(k 1 ) and to v 2 ∈ Spv(k 2 ). Suppose first that k 1 /k is a finite extension. By Lemma 2.7, we are free to first replace k 1 and k 2 by suitable algebraic extensions; we may thus ensure that k 1 is normal over k and that k 2 contains a subfield k ′ 1 isomorphic to k 1 . In this case, Lemma 2.7 implies the existence of an isomorphism k 1 ∼ = k ′ 1 compatible with valuations, proving the claim.
To check the general case, by Zorn's lemma, we may assume that k 2 = k(x). By Lemma 2.7 and the previous paragraph, we may further assume that both k and k 1 are algebraically closed; by adjoining an extra transcendental, we may further assume that the valuations on k, k 1 , k 2 are all nontrivial. By quantifier elimination in ACVF (e.g., see [12]), for any rational functions f 1 , . . . , f n ∈ k(T ), there exist a field extension k 3 of k 1 , a Krull extension v 3 on k 3 restricting to v 1 on k 1 , and an element y ∈ k 3 such that for i = 1, . . . , n, we have v 2 (f i (x)) ≤ 1 if and only if v 3 (f i (y)) ≤ 1.
By Lemma 2.6, Spv(k 1 (x)) is a spectral space and hence is compact for the patch topology. By the previous paragraph and the finite intersection property, there exists v 3 ∈ Spv(k 1 (x)) restricting to v 1 on k 1 and to v 2 on k(x) ∼ = k 2 . This proves the claim. Remark 2.10. As one may infer from the analogy with schemes, the map in Lemma 2.8(b) is not injective. For example, if k 1 = k(x), k 2 = k(y) with x, y transcendental over k, then the set Spv(k 1 ⊗ k k 2 ) contains the trivial valuation on k 1 ⊗ k k 2 , but it also contains many nontrivial semivaluations which restrict trivially to k 1 , k 2 . One of these may be constructed by restricting the trivial valuation along the map k 1 ⊗ k k 2 → k 1 which acts on k 1 as the identity map and on k 2 as the k-linear identification k 2 ∼ = k 1 mapping y to x.

Ordinary adic spectra
We next recall Huber's construction of adic spectra and definition of adic spaces.
Definition 3.1. A linearly topologized ring (or LT ring for short) is a topological ring A admitting a neighborhood basis of 0 consisting of additive subgroups. For A an LT ring, a subset B of A is bounded if for each neighborhood U of 0 in A, there exists a neighborhood V of 0 in A with V ·B ⊆ U . An element a ∈ A is power-bounded (resp. topologically nilpotent) if the sequence a, a 2 , . . . is bounded (resp. converges to 0). The set A • of powerbounded elements is a subring of A; the set A •• of topologically nilpotent elements is an ideal of A • .   itself, and so we may as well take f 0 = 1.
Remark 3.5. The definition of a rational subspace of Spv(A) we are using is the one from [18]. The definition in [17] is formally different, but again can be shown to lead to the same class of subspaces.
Definition 3.6. An adic ring is a topological ring A admitting an ideal I whose powers form a fundamental system of neighborhoods of 0. Any ideal with this property is called an ideal of definition of A. An f-adic ring is a topological ring A admitting an open subring A 0 which is adic with a finitely generated ideal of definition. Any such subring A 0 is called a ring of definition of A. Note that any f-adic ring is LT, and the tensor product of f-adic rings (in the sense of Definition 3.2) is again f-adic. 7. An f-adic ring A is Tate if it contains a topologically nilpotent unit. In this case, any open ideal is trivial; that is, if f 1 , . . . , f n generate an open ideal of A, then for any v ∈ Spv(A), the quantities v(f 1 ), . . . , v(f n ) cannot all vanish (e.g., see Corollary 4.13 below). One consequence of this is that (3.4) can be rewritten as This modification is needed to compare the concept of a rational subspace of Spv(A) with analogous concepts, such as that of a rational subspace of an affinoid space in rigid analytic geometry (as in [3]).     Proof. See [16,Theorem 3.5(i,ii)]. The first assertion can also be deduced from Theorem 3.9 using Corollary 1.14.      (a) The morphism Definition 3.18. A locally valuation-ringed space, or locally v-ringed space for short, is a locally ringed space (X, O X ) equipped with the additional data of, for each x ∈ X, a valuation v x on the local ring O X,x . A morphism of locally v-ringed spaces f : X → Y is a morphism of locally ringed spaces with the property that for each x ∈ X mapping to y ∈ Y , the restriction of v x along the map O Y,y → O X,x is equal to v y . We say that (A ⊲ , A + ) is sheafy if the presheaf O is in fact a sheaf; in particular, A ⊲ must be complete. In this case, (X, O) is a locally ringed space, which we promote to a locally v-ringed space as follows: for x ∈ X corresponding to v ∈ Spv(A), let v x be the continuous extension of v to O X,x . Any locally v-ringed space of this form is called an affinoid adic space. A locally v-ringed space which is covered by open subspaces which are affinoid adic spaces is called an adic space.
Unfortunately, the sheafy condition is not always satisfied; see [5,22] for counterexamples. Two important classes where it is satisfied are described by the following results of Huber (in case (a)) and Buzzard-Verberkmoes (in case (b)).
Theorem 3.20. Suppose that A ⊲ is Tate and that at least one of the following conditions holds.
(a) The ring A ⊲ is strongly noetherian: for each nonnegative integer n, Remark 3.21. There is a process to attach "spaces" to affinoid f-adic rings which are not sheafy, but this requires a more abstract approach as originally described by Scholze and Weinstein [24]. See also [20, §8.2].
Remark 3.22. Huber declares an adic space to be analytic if it is covered by the adic spectra of affinoid f-adic rings which are not only sheafy, but also Tate. This extra restriction fails in some natural classes of examples (e.g., adic spaces associated to ordinary schemes or formal schemes), but is needed in order to make many classical arguments of rigid analytic geometry carry over to the setting of adic spaces. One pleasant feature of reified adic spaces is that there admit no analogue of the analytic condition; the role played by topologically nilpotent units is taken over by reifications.

Gel'fand spectra
We next introduce the class of normed rings and describe Berkovich's construction of the Gel'fand spectrum of a normed ring.
(c) For all x, y ∈ A, we have |xy| ≤ |x||y|. We say that a (semi)normed ring A is nonarchimedean if the upper bound in (b) can be improved to max{|x|, |y|}. The trivial norm on A is the norm for which |x| = 1 for all nonzero x ∈ A.
The (semi)norm topology on a nonarchimedean (semi)normed ring A is the metric topology induced by the seminorm. For this topology, A is an LT ring.
Any such morphism is continuous (but not conversely).

Definition 4.3.
A (nonarchimedean commutative) Banach ring is a nonarchimedean normed ring which is separated and complete for the norm topology. For A a Banach ring, a Banach algebra over A is a Banach ring B equipped with a bounded homomorphism A → B.

Definition 4.4.
A ultrametric field is a Banach ring F such that the underlying ring F is a field and the norm is a Krull valuation (i.e., the inequality in (c) is an equality). Unless otherwise specified, we allow this definition to include the case of a trivial norm.
Remark 4.5. In Definition 4.4, the second condition is needed because one can modify the norm on F without changing the norm topology, in such a way that the resulting norm is not itself a Krull valuation, e.g., by taking the supremum of the norms corresponding to two different reifications of the same underlying valuation. (Compare [19,Remark 8.7].) Remark 4.6. Any f-adic ring A can be viewed as a nonarchimedean seminormed ring (topologized using the seminorm topology). For example, let A 0 be a ring of definition, let I be a finitely generated ideal of definition of A 0 , pick c ∈ (0, 1), and define |•| : A → [0, +∞) as follows.
• For a ∈ A 0 , set |a| = c −n for n the smallest nonnegative integer such that a / ∈ I n+1 if such an integer exists; otherwise, set |a| = 0. • For a / ∈ A 0 , set |a| = c n for n the smallest positive integer such that aI n ⊆ A 0 . Such an integer must exist because A 0 is open in A. Beware that the equivalence class of this norm is not uniquely determined by the topology of A (because of the possibility of varying c and I); in particular, this construction does not define a functor from f-adic rings to nonarchimedean seminormed rings.
In the other direction, for A a nonarchimedean seminormed ring viewed as an LT ring using the seminorm topology, it is not immediate that A is an f-adic ring; the difficulty is to find an ideal of definition which is finitely generated. One case where this is possible is when A contains a topologically nilpotent unit x (i.e., A is Tate), by taking A 0 to be the subring of x ∈ A for which |x| ≤ 1 and I to be the ideal (x n ) for n large enough so that x n ∈ A 0 ; consequently, any such A is a Tate f-adic ring. In particular, if A is a nonzero Banach algebra over an ultrametric field F with nontrivial norm, any x ∈ F with 0 < |x| < 1 is a topologically nilpotent unit.
As remarked above, an f-adic ring cannot be viewed as a nonarchimedean seminormed ring in a canonical way. However, we have the following result.
Lemma 4.7. Let R be a Banach ring which is Tate. Then the forgetful functor from Banach rings over R to complete f-adic rings A equipped with continuous maps R → A is an equivalence of categories.
Proof. By Remark 4.6, the functor is essentially surjective; full faithfulness is a consequence of the Banach open mapping theorem (see [13]).
Definition 4.8. For A → B, A → C two bounded homomorphisms of nonarchimedean seminormed rings, we view the tensor product B ⊗ A C as a nonarchimedean seminormed ring by equipping it with the tensor product seminorm: the value of the seminorm on Definition 4.10. Let α be an R-valued semivaluation on A. We may then extend α to an R-valued Krull valuation on Frac(A/ ker(α)). Completing with respect to this extension yields an ultrametric field, denoted H(α). Note that α can be recovered as the restriction along the natural map A → H(α) of the valuation on H(α).
For the remainder of §4, let A be a nonarchimedean normed ring. Proof. If I does not contain 1 in its closure, then the quotient seminorm on A/I is nonzero, so Theorem 4.12 applies to produce α ∈ M(A) whose restriction to I is zero.
. . , f n generate the unit ideal and some q 1 , . . . , q n > 0. If it is possible to take q 1 = · · · = q n = 1, we call the resulting set a strictly rational subspace of M(A). As in Definition 3.3, the intersection of two (strictly) rational subspaces is (strictly) rational: taking q 0 = r 0 = 1, we have As in Definition 3.7, by Corollary 4.13 the space (4.15) can also be written as Hence any rational subspace of M(A) is closed. Remark 4.17. With notation as in Definition 4.14, note that by compactness, . . , f ′ n again generate the unit ideal in A, and

Reified valuations
In order to bring the valuation-theoretic and norm-theoretic viewpoints into alignment, and to give an explicit relationship between the two in the case of affinoid algebras (Theorems 9.5 and 9.6), we describe a variation on the theory of valuations in which scaling ambiguities are eliminated. Much of the resulting analysis runs parallel to the analysis in [16] cited above, although with some key differences due to the change in the definition of rational subspaces (see Remark 4.16).
Definition 5.1. Let R + denote the multiplicative monoid of positive real numbers. A reified value group is a value group Γ equipped with an orderpreserving homomorphism r : R + → Γ.
Let A be a ring. A reified semivaluation on A is a semivaluation v : A → Γ 0 for Γ a reified value group. Given a semivaluation v : A → Γ 0 , we will refer to the extra data of an order-preserving homomorphism r : For v a reified semivaluation, let Γ v be the subgroup of Γ generated by R + and the nonzero images of A, viewed as a reified value group. Two reified Definition 5.2. Let A be a ring. The reified valuative spectrum of A, denoted Sprv(A), is the set of equivalence classes of reified semivaluations on A, equipped with the topology generated by sets of the form Again, if we let B be the Boolean algebra generated by the basic open sets as in (5.3), then B is also generated by the sets of the form Remark 5.5. There is a natural projection Sprv(A) → Spv(A) forgetting reifications, which is surjective for trivial reasons: given any semivaluation v : A → Γ 0 , we may form an equivalent semivaluation by enlarging the value group to Γ × R + (or R + × Γ) ordered lexicographically. For a more refined statement along the same lines, see Lemma 5.9.
The analogue of the fact that the equivalence class of a valuation is determined by its order relation is the following.
Then this map is injective and its image is cut out by the following closed conditions (writing a, b, c, d for arbitrary elements of A and q, r for arbitrary elements of R + ): Proof. Each condition is evidently closed and satisfied on the image of Sprv(A). Conversely, suppose that the tuple (v a,b,q ) belongs to the image. We reconstruct the corresponding reified valuation v as follows.
We next reconstruct the reified divisibility relation.
By (i), the relation ≤ is reflexive. We next check that for (a, b, q) ∈ S, On one hand, if a ∈ p v , then ad ∈ p v and so v 0,ad,1 = 1; by (ix), v bc,0,q/r = 1; by (iii), we have (a, b, q) ≤ (c, d, r). On the other hand, if (a, b, q) ≤ (c, d, r) and c ∈ p v , then v bc,ad,q/r = 1 and bc ∈ p v , so by (iii), v 0,ad,q/r = 1. By (ix), v 0,1,r/q = 1; by (iv), ad ∈ p v . Since p v is prime, a ∈ p v .
To check that ≤ is transitive, we assume (a, b, q) ≤ (c, d, r) ≤ (e, f, s) and distinguish two cases. If c ∈ p v , then a ∈ p v by (5.7) and so (a, b, q) ≤ (e, f, s). If c / ∈ p v , then v bc,ad,q/r = 1 and v de,cf,r/s = 1, so by (iv), v bcde,acdf,q/s = 1. Since cd / ∈ p v , by (viii) we have (a, b, q) ≤ (e, f, s). We next reconstruct the underlying reified value group. Define an equivalence relation equating (a, b, q), (c, d, r) ∈ S whenever (a, b, q) ≤ (c, d, r) and (c, d, r) ≤ (a, b, q). Let Γ v,0 be the set of equivalence classes. Let 0 ∈ Γ v,0 be the class of (0, 1, 1); by (5.7), this consists of those (a, b, q) with a ∈ p v . Equip S with the binary operation · given by (a, b, q) · (c, d, r) → (ac, bd, qr); for this operation, S is a commutative monoid with identity element (1, 1, 1). By (iv), · is monotonic with respect to ≤; it thus induces a monoid structure on S. For (a, b, q) ∈ S with a / ∈ p v , we have ab / ∈ p v and so (b, a, 1/q) ∈ S; by (i), (a, b, q) and (b, a, 1/q) define inverse classes in Γ v,0 . If we set Γ v = Γ v,0 \ {0} and define the map r : R + → Γ v taking q to the class of (1, 1, q), it follows that Γ v is a reified value group with identity element the class of (1, 1, 1) and associated pointed commutative monoid Γ v,0 .
To conclude, let v : A → Γ v,0 be the function taking a to the class of (a, 1, 1). By (vii) and (iv), v is a reified semivaluation whose image in This gives rise to the following analogue of Lemma 2.6, with a similar proof.
is surjective (but typically not injective).
Proof. We may reduce to the case where ℓ = k(x) for some x ∈ ℓ. Let v 1 be a valuation on ℓ restricting to the valuation v on k. If the inclusion then any reification of v induces a unique reification of v 1 . This already suffices to treat the case where x is algebraic over k.
If x is transcendental over k, using the previous paragraph we may reduce to the case where k is itself algebraically closed and Γ v 1 = Γ v (the latter group being divisible). Since k is algebraically closed, there must exist since the nonzero terms in the maximum are pairwise distinct. It follows that as abstract groups we have If v is trivial, then the claim is equally trivial, so we may assume hereafter that v is nontrivial. Letṽ be a reification of v. Let S − (resp. S + ) be the set of t ∈ R for which there exists y ∈ k such that v 1 (cx − d) > v 1 (y) andṽ(y) ≥ t (resp. v 1 (cx − d) < v 1 (y) andṽ(y) ≤ t). These sets have the following properties.
• The set S − is down-closed and contains 0.
• The set S + is up-closed and contains 1.
• The sets S − , S + are disjoint.
• The supremum of S − equals the infimum of S + . We denote the common value by s. Note that it is possible to choose ǫ ∈ {−1, 0, 1} subject to the following conditions.
• If s ∈ {0, 1}, then ǫ = 0. Let Γ be the lexicographic product Γṽ × R + viewed as a reified value group via the reification on the first factor. We then obtain an embedding of Γ v 1 into Γ taking v 1 (cx − d) to (s, e ǫ ), yielding a reification of v 1 as desired.
We have the following analogue of Lemma 2.8.
Proof. This is immediate from Lemma 2.8 and Lemma 5.9.
For the remainder of §5, fix a nonarchimedean normed ring A. We first cut down the space Sprv(A) by imposing some interaction between the seminorm on A and the reifications.
Definition 5.11. For r ∈ R + , let A •,r (resp. A ••,r ) be the set of a ∈ A such that the sequence {r −n |a n |} ∞ n=1 is bounded (resp. converges to 0). Note that A •,1 = A • and A ••,1 = A •• ; more generally, if a ∈ A •,r , then |a| sp ≤ r, but not conversely. By contrast, if a ∈ A ••,r then |a| sp < r and conversely: if |a| sp < r, then there exist a positive integer m and a value c ∈ (0, 1) such that |a m | ≤ c m r m , so for all n ≥ 0 we have r −n |a n | ≤ c ⌊n/m⌋ max{r −i a i : i = 0, . . . , m − 1} and so a ∈ A ••,r . Following Temkin [26], we define the graded ring The analogue of continuity for reified valuations is the following condition. This construction is designed to eliminate certain infinitesimals.
for some f 0 , . . . , f n ∈ A such that f 1 , . . . , f n generate the unit ideal and some q 1 , . . . , q n > 0. As in Definition 4.14, we can rewrite (5.19) as Any rational subspace is quasicompact (by Lemma 5. Proof. Choose any v ∈ Comm(A). For any a, b ∈ A and q > 0, if the set contains v, then there exists r ∈ R + such that r ≤ v(b), so is a rational subspace of Comm(A) containing v and contained in U . Since the set of rational subspaces is closed under finite intersections, this proves the claim.
The following result is analogous to [16,Proposition 2.6], although the proof is somewhat different. Since rational subspaces of Comm(A) are quasicompact, by Lemma 5.21 the patch topology on Comm(A) is generated by rational subspaces and their complements. By this remark plus Lemma 5.22, r is continuous for the patch topologies, so Comm(A) is compact for the patch topology. By Corollary 1.11, Comm(A) is spectral. Since r is continuous for the patch topologies, Corollary 1.10 implies that r is spectral.
Remark 5.24. In Huber's theory, including valuations which are not continuous would give rise to spaces which detect nontrivial blowups, i.e., analogues of the Riemann-Zariski spaces associated to schemes. For instance, for any two f, g ∈ A, some points at which both f and g vanish would be separated by assigning a nonzero but infinitesimal value to f /g. A similar effect would be achieved here by including reified valuations which are not commensurable; this point of view is taken in [7].

Reified adic spectra
We now define an intermediate construction between adic spectra and Gel'fand spectra, starting with an analogue of the definition of an affinoid f-adic ring.
Definition 6.1. By an affinoid seminormed ring, we will mean a pair (A ⊲ , A Gr ) in which A ⊲ is a nonarchimedean seminormed ring and A Gr is an integrally closed graded subring of Gr A ⊲ . If A ⊲ is separated and complete for its seminorm, we also call such a pair an affinoid Banach ring. For r ∈ R + , write A +,r for the subring of A ⊲,•,r whose image in Gr r A ⊲ belongs to A Gr,r .
A   Proof. Immediate from Theorem 4.12.
We have the following analogue of Theorem 3.14.
is surjective.
Proof. Given v 1 ∈ Spra(B ⊲ , B Gr ), v 2 ∈ Spra(C ⊲ , C Gr ) mapping to v ∈ Spra(A ⊲ , A Gr ), Lemma 5.10 produces v ∈ Sprv(D ⊲ ) restricting to v 1 ∈ Spra(B ⊲ ) and to v 2 ∈ Sprv(C ⊲ ). By the construction of D Gr , it is automatic that for any r ∈ R + and any a ∈ D +,r , we have v(a) ≤ r. In particular, we have v ∈ Sprb(D ⊲ ). By contrast, it is not automatic that v is commensurable, but we may enforce this by applying the map r of Definition 5.17.
Recall that Definition 5.15 gives rise to a projection map Spra(A ⊲ , A Gr ) → M(A ⊲ ). The fibers of this map may be described as follows.
Definition 6.6. For F an ultrametric field, a graded valuation ring of F is a graded subring R of Gr F with the property that for each r > 0 and each nonzero a ∈ Gr r F , either a or a −1 (or both) belongs to R. In particular, the graded piece R 1 of R is a valuation ring in the residue field Gr 1 F of F . Lemma 6.7. Let (F ⊲ , F Gr ) be an affinoid seminormed ring such that F ⊲ is an ultrametric field. Then there is a natural bijection between Spra(F ⊲ , F Gr ) and the set of graded valuation rings of F ⊲ containing F Gr .
Proof. Given v ∈ Spra(F ⊲ , F Gr ), we may construct a graded valuation ring R v of F ⊲ containing F Gr as follows: for a ∈ F with |a| ≤ r, the class of a in Gr r F ⊲ belongs to R r v if and only if v(a) ≤ r. (To see that R v is indeed a graded valuation ring, note that if a ∈ F satisfies |a| ≤ r but the class of a does not belong to R r v , then |a| = r and v(a) > r, so a −1 = r −1 and v(a −1 ) < r −1 .) In the other direction, given a graded valuation ring R of F ⊲ containing F Gr , for a, b ∈ F ⊲ and q > 0, put  Remark 6.11. Conversely, let (A ⊲ , A + ) be an affinoid f-adic ring for which A ⊲ has been equipped with the structure of a nonarchimedean seminormed ring (this is always possible but not canonical; see Remark 4.6). We may then form an affinoid seminormed ring by viewing A + /A •• as a graded subring of Gr A ⊲ concentrated in Gr 1 A ⊲ . Note that applying Remark 6.10 then recovers A + . Remark 6.12. In Remark 6.11, we may apply Lemma 5.9 to see that the projection map Spra(A ⊲ , A + ) → Spa(A ⊲ , A + ) is surjective; however, this map does not in general admits a distinguished section. One case where this does occur is when A ⊲ is a Banach algebra over an ultrametric field F with norm group R + : the canonical reification of the norm on F fixes a reification on every semivaluation. For a related argument, see the proof of Lemma 7.2.
7. The structure presheaf on a readic spectrum With Huber's adic spaces as a model, we now introduce the structure presheaf on a reified adic spectrum and build the reified analogues of adic spaces. This development parallels the corresponding foundations in Huber's theory, for which we follow [20, §2.4] and sources cited therein. However, as indicated in Remark 3.22, the role of the Tate condition is largely eliminated by the presence of reifications.

Proof. It is clear that if
Conversely, suppose that x / ∈ A +,r . Let B ⊲ be the completion of the group ring A ⊲ [R + ] for the norm taking r∈R + a r [r] to max{|a r |r}. Extend B ⊲ to an affinoid f-adic ring (B ⊲ , B + ) with B + = ⊕ s∈R + A +,s [s −1 ]. By construction, x[r −1 ] / ∈ B + , so we may apply [16, Lemma 3.3] to produce w ∈ Spa(B ⊲ , B + ) such that w(x[r −1 ]) > 1. Let v : A ⊲ → Γ w,0 be the restriction of w, viewed as a reified valuation for the map s → w([s]); then v ∈ Spra(A ⊲ , A Gr ) and v(x) > r, as desired.
It is clear that every rational subspace of X contained in U pulls back to a rational subspace of Spra(B ⊲ , B Gr ). Conversely, given a rational subspace V of Spra(B ⊲ , B Gr ) defined by some parameters g 0 , . . . , g m ∈ B ⊲ and some scale factors r 1 , . . . , r m > 0, by Remark 4.17 we may choose the g i to be in the image of A ⊲ [T 1 , . . . , T n ]. By multiplying through by a suitable power of f 0 , we obtain parameters in A ⊲ itself, but these parameters need not generate the unit ideal in A ⊲ . However, since α(g 0 ) = 0 for all α ∈ M(B ⊲ ), by compactness we can find c > 0 such that α(g 0 ) ≥ c for all α ∈ M(B ⊲ ). Put g m+1 = 1 and r m+1 = c −1 ; then the parameters g 0 , . . . , g m+1 and scale factors r 1 , . . . , r m+1 define a rational subspace of X whose intersection with U corresponds to V . Since the intersection of two rational subspaces is again a rational subspace, this completes the proof of (b). As in classical rigid geometry, most of our arguments about the structure presheaf involve a reduction to certain special types of coverings.  For f 1 , . . . , f n ∈ A ⊲ generating the unit ideal and q 1 , . . . , q n > 0, the standard rational covering of X generated by f 1 , . . . , f n with scale factors q 1 , . . . , q n is the covering of X by the rational subspaces For f 1 , . . . , f n ∈ A arbitrary and q 1 , . . . , q n > 0, the standard Laurent covering generated by f 1 , . . . , f n with scale factors q 1 , . . . , q n is the covering by the rational subspaces A standard Laurent covering with n = 1 is also called a simple Laurent covering.
Lemma 7.9. The following statements hold.
(a) Any finite covering of X by rational subspaces can be refined by a standard rational covering. (b) For any standard rational covering U of X, there exists a standard Laurent covering V of X such that for each V = Spra(B ⊲ , B Gr ) ∈ V, the restriction of U to V (omitting empty intersections) is a standard rational covering generated by units in B ⊲ . (c) Any standard rational covering of X generated by units can be refined by a standard Laurent covering generated by units.
Proof. To prove (a), we follow [3, Lemma 8.2.2/2]. We start with a finite covering of X by rational subspaces U 1 , . . . , U n , where U i is generated by the parameter set S i = {f i0 , f i1 , . . . , f in i } with corresponding scale factors q i1 , . . . , q in i . Let S be the set of products of the form s 1 · · · s n where s i ∈ S i for all i. Let S ′ be the subset of S consisting of products s 1 · · · s n for which s i = f i0 for at least one i. Note that S ′ generates the unit ideal: for any v ∈ X, for each i we can find s i ∈ S i not vanishing at v, taking s i = f i0 for any i for which v ∈ U i . Thus the parameter set S ′ can be used to define a standard rational covering; we do so by taking the scale factor associated to f 1j 1 · · · f njn to be q 1j 1 · · · q njn . To see that this refines the original covering, note that the rational subspace with first parameter s 1 · · · s n does not change if we add S \ S ′ to the set of parameters (again because the U i form a covering), which makes it clear that this subspace is contained in U i for any index i for which s i = f i0 (because we now have parameters obtained from s 1 , . . . , s n by replacing s i with each of the other elements of S i ).
To prove (b), we follow [3, Lemma 8.2.2/3]. Let U be the standard rational covering defined by the parameters f 1 , . . . , f n with scale factors q 1 , . . . , q n . Since f 1 , . . . , f n generate the unit ideal, by Corollary 4.13 the quantity is positive. In this case, the standard Laurent covering V defined by f 1 , . . . , f n with scale factors c/2, . . . , c/2 has the desired property: on the subspace where q j |f j | ≤ c/2 for j = 1, . . . , s and q i |f i | ≥ c/2 for i = s + 1, . . . , n, the restriction of U is the standard rational covering generated by f s+1 , . . . , f n with scale factors q s+1 , . . . , q n plus some empty intersections.
To prove (c), we follow [3,Lemma 8.2.2/4]. Consider the standard rational covering generated by the units f 1 , . . . , f n with scale factors q 1 , . . . , q n . This cover is refined by the standard Laurent covering generated by f i f −1 j with scale factors q i /q j for 1 ≤ i < j ≤ n, by an elementary combinatorics argument (any total ordering on a finite set has a maximal element).
This yields the following reduction argument, analogous to [20,Lemma 2.4.19].
Lemma 7.10. Let P be a property of finite coverings of rational subspaces of X by rational subspaces. Suppose that P satisfies the following condition. Then the property P holds for any finite covering of a rational subspace of X by rational subspaces.
This yields the following criterion for sheafiness and acyclicity.
Lemma 7.11. Let F be a presheaf of abelian groups on X such that for every rational subspace U of X and every simple Laurent covering V 1 , V 2 of U , we have Then for every rational subspace U of X and every finite covering V of U by rational subspaces, Proof. This follows from Lemma 7.10 as in [20,Proposition 2.4.21].
Using this criterion, we may see that sheafiness implies acyclicity, by analogy with [20,Theorem 2.4.23].
Lemma 7.13. Let S − , S + be the simple Laurent covering of X defined by some f ∈ A ⊲ and some q > 0. Let ) be the morphisms representing the rational subspaces S − , S + , S − ∩ S + of X.
Proof. By Lemma 7.4, we obtain strict surjections taking T to f and U to f −1 . In particular, any b ∈ B ⊲ 12 can be lifted to some Let a ′ n be the sum of a ij over all i, j ≥ 0 with i − j = n; note that this sum converges in , proving the desired exactness. Theorem 7.14. Suppose that (A ⊲ , A Gr ) is sheafy. Then for every finite covering U of X by rational subspaces, Proof. By Lemma 7.11, it suffices to checkČech-acyclicity for simple Laurent coverings. Since the sheafy condition propagates to rational subspaces, we may as well consider only simple Laurent coverings of X itself. In the notation of Lemma 7.13, the sequence ; by the sheafy hypothesis, it is also exact at We may also establish a weak analogue of Kiehl's theorem on coherent sheaves, by analogy with [20, Theorem 2.7.7]. Proof. We may use Lemma 7.10 to reduce to the case where notation is as in Lemma 7.13 and one is given a sheaf whose restrictions to S − , S + , S − ∩ S + correspond to finite projective modules M 1 , M 2 , M 12 over B ⊲ 1 , B ⊲ 2 , B ⊲ 12 , respectively. In this case, by Theorem 7.14, the diagram Definition 7.17. A locally reified valuation-ringed space, or locally rv-ringed space for short, is a locally ringed space (X, O X ) equipped with the additional data of, for each x ∈ X, a reified valuation v x on the local ring O X,x . A morphism of locally rv-ringed spaces f : X → Y is a morphism of locally ringed spaces with the property that for each x ∈ X mapping to y ∈ Y , the restriction of v x along the map O Y,y → O X,x is equal to v y as a reified valuation.
Definition 7.18. Any locally rv-ringed space of the form Spra(A ⊲ , A Gr ) for some sheafy (A ⊲ , A Gr ) is called an affinoid reified adic space. For such a space, we recover A ⊲ as the ring of global sections; by Lemma 7.2, we may recover A Gr from the reified valuations on local rings. A locally v-ringed space which is covered by open subspaces which are affinoid reified adic spaces is called a reified adic space. We suggest to abbreviate reified adic space to readic space or R-adic space. As in Remark 3.21, one can formally define a "space" associated to a nonsheafy (A ⊲ , A Gr ) using a functor of points approach, by analogy with [20, §8.2].

On the sheafy condition
By analogy with Theorem 3.20, we identify two important classes of sheafy affinoid seminormed rings. In the analogue of the strongly noetherian case, we also get a more precise analogue of Kiehl's characterization on coherent sheaves on affinoid spaces. We begin with the analogue of the stably uniform condition.
Definition 8.1. We say that a Banach ring A is uniform if its norm is equivalent to its spectral seminorm. (An equivalent condition is that there exists c > 0 such that for all a ∈ A ⊲ , a 2 ≥ c |a| 2 .) We say that an affinoid Banach ring (A ⊲ , A Gr ) is really stably uniform if for any homomorphism (A ⊲ , A Gr ) → (B ⊲ , B Gr ) representing a rational subspace of X, B ⊲ is uniform. See [5] for some exotic examples related to these conditions (e.g., for uniform rings which are not really stably uniform).
Lemma 8.2. Let A be a uniform Banach ring. For any f ∈ A, the ideals are closed.
By analogy with Theorem 3.20(b), we have the following.
is really stably uniform, then it is sheafy.
Proof. By Lemma 7.11 and Lemma 7.13, it suffices to check that with notation as in Lemma 7.13, the sequence In particular, if a ∈ ker(A ⊲ → B ⊲ 1 ⊕ B ⊲ 2 ), then then a has zero spectral seminorm; however, since A is uniform by hypothesis, this forces a = 0.
We check exactness at In the commutative diagram the first two rows are clearly exact, while the columns are exact by Lemma 8.2. By diagram chasing, we obtain exactness of the third row at B ⊲ 1 ⊕ B ⊲ 2 . We next turn to the analogue of the strongly noetherian condition, where we can carry out a more thorough adaptation of Huber's constructions.
Definition 8.4. We say that a Banach ring A is really strongly noetherian if A ⊲ {T 1 /r 1 , . . . , T n /r n } is noetherian for all n ≥ 0 and all r 1 , . . . , r n > 0. We say that an affinoid Banach ring (A ⊲ , A Gr ) is really strongly noetherian if A ⊲ is really strongly noetherian; this implies that for every morphism (A ⊲ , A Gr ) → (B ⊲ , B Gr ) representing a rational subspace of Spra(A ⊲ , A Gr ), B ⊲ is really strongly noetherian.
Example 8.5. Any ultrametric field is really strongly noetherian by [2, Proposition 2.1.3], as then is any Berkovich affinoid algebra over an ultrametric field (see Definition 9.4).
Example 8.6. Let F be a ultrametric field with nontrivial norm which is perfect of characteristic p. Let W (F ) be the ring of p-typical Witt vectors of F , which may be viewed as the unique p-adically separated and complete ring whose reduction modulo p is F . Each x ∈ W (F ) can be written uniquely as ∞ n=0 p n [x n ] with x n ∈ F and brackets denoting Teichmüller lifts. The set of x ∈ W (F ) for which lim n→∞ p −n |x n | = 0 is then a really strongly noetherian Banach ring for the norm x → max n {p −n |x n |}; see [19,Theorem 3.2].
We mention one further class of examples. Theorem 8.7. Let A be a noetherian ring equipped with the trivial norm. Then A is really strongly noetherian.
Proof. As in [19,Theorem 3.2], we use a Gröbner basis construction. Choose r 1 , . . . , r n > 0. Equip Z n ≥0 with the componentwise partial order ≤, and with the graded lexicographic total ordering . Since ≤ is a well-quasi-ordering (every sequence contains an infinite ascending subsequence) and refines ≤, is a well-ordering. For x = I x I T I ∈ A{T 1 /r 1 , . . . , T n /r n } nonzero, consider those indices I for which x I = 0 and r i 1 1 · · · r in n is maximized, then identify the greatest such index with respect to ; we define the leading index and leading coefficient of x to be the resulting values of I and x I , respectively.
For J an ideal of A{T 1 /r 1 , . . . , T n /r n } and I ∈ Z n ≥0 , let L I be the ideal of A consisting of 0 plus the leading coefficients of all elements of J with leading index I. For I 1 ≤ I 2 we have L I 1 ⊆ L I 2 . Let S be the set of indices I for which L I = L I ′ for any I ′ < I; this set is finite by the well-quasiordering property of ≤ and the noetherian property of A. For each I ∈ S, let G I be a set of elements of J realizing each leading coefficient in some finite set of generators of L I . We may then present each element x ∈ J as a linear combination of elements of ∪ I∈S G I by repeatedly applying the usual division algorithm as long as x = 0: identify the leading index of x as a multiple of some element I of S, then kill off the leading coefficient of x by subtracting off a suitable monomial linear combination of elements of G I . Corollary 8.8. Let A be the ring Z((z)) equipped with the z-adic norm (for any normalization). Then A is really strongly noetherian.
One important consequence of the really strongly noetherian condition is that it allows topological considerations to be omitted from many algebraic constructions involving finitely generated modules. Lemma 8.9. Let A be a really strongly noetherian Banach ring.
(a) Every ideal in A is closed.
(b) Every finite A-module is complete under the quotient topology induced by some (and hence any) surjection from a finite free module. (c) Every morphism between finite A-modules, topologized as in (b), is strict.
Proof. Suppose first that A is Tate. We first observe that if M is a normed A-module whose completion M is finitely generated, then M = M . This is proved as in [3, Proposition 3.7.3/2]: choose an A-linear surjection f : A n → M , apply the Banach open mapping theorem for A (see [13]) to deduce that f is strict, then conclude by Nakamaya's lemma in the form of [3, Lemma 1.2.4/6].
We now check that for any finite free A-module F , any submodule M of F is complete. To wit, choose any r ∈ (0, 1) and put B = A{T /r, U/r −1 }/(T U − 1); then B is necessarily Tate. By the previous paragraph, the image of As a consequence, we obtain some results on flatness of certain ring homomorphisms. Proof. By induction, we reduce to the case n = 1 and put T = T 1 , r = r 1 . To handle this case, we follow [18,Lemma 1.7.6].
We first prove that A → A{T /r} is flat. For M a finite A-module, by Lemma 8.9, M is complete for its natural topology and any finite presentation of M is strict. We may thus identify (applying Lemma 8.9 to obtain the last isomorphism). Since the ordinary lo- This proves (a), from which (b) follows by invoking Lemma 7.5 and some standard commutative algebra (see for instance [25,Tag 00HQ]).
Corollary 8.12. Let (A ⊲ , A Gr ) be a really strongly noetherian affinoid Banach ring. Then for every rational subspace U of X and every finite covering V of U by rational subspaces, the maps are injective. (They will be shown to be bijective in Theorem 8.15.) Proof. Immediate from Corollary 8.11.
We mention also a refinement of Corollary 8.11, which gives a stronger result but has a somewhat mysterious extra hypothesis.   Proof. Again, we may use Lemma 7.10 to reduce to the case where notation is as in Lemma 7.13 and and one is given a sheaf F whose restrictions to are exact; by the really strongly noetherian hypothesis, N * is a finitely generated B ⊲ * -module. By Corollary 8.11, the induced maps N i ⊗ B ⊲ i B ⊲ 12 → N 12 are isomorphisms. We may thus repeat the previous argument to see that G is globally finitely generated; that is, there exists an exact sequence of the form We may now take global sections to obtain a finite A ⊲ -module coker(A ⊲⊕m → A ⊲⊕n ) whose associated sheaf is isomorphic (by the right exactness of tensor products) to F.  [20], while the special case where A ⊲ is really strongly noetherian follows from Theorem 8.16. Some additional results in this direction can be found in [9].

Comparison of Grothendieck topologies
We now study the relationship between Gel'fand spectra and readic spectra, following the study of the relationship between rigid and adic spaces made by Huber [16, §4] and van der Put and Schneider [27].
Definition 9.1. For A a nonarchimedean normed ring, define the strictly special G-topology (resp. the special G-topology) on M(A) by taking the admissible open subsets to be the finite unions of strictly rational subspaces (resp. rational subspaces) of M(A) and taking the admissible coverings to be the finite set-theoretic coverings. Both G-topologies are prespectral; the special G-topology is also T 0 . In general, the maps i, j do not hit enough points of Spra(A) or Spa(A) to make it possible to recover the structure of these spaces from M(A ⊲ ). One crucial exception is the case of classical affinoid algebras. For the remainder of §9, let F be an ultrametric field. Definition 9.4. A Tate affinoid algebra over F is a Banach algebra A over F which can be realized as a topological quotient of F {T 1 , . . . , T n } for some n. If the norm on F is nontrivial, then every maximal ideal of A has residue field finite over F [3, Corollary 6.1.2/3], so we obtain a natural inclusion Maxspec(A) → M(A).
A Berkovich affinoid algebra over F is a Banach algebra over F which can be realized as a topological quotient of F {T 1 /r 1 , . . . , T n /r n } for some n and some r 1 , . . . , r n > 0.
Theorem 9.5. Assume that the norm on F is nontrivial, and let A be a reduced Tate affinoid algebra over F . To prove (c), we treat only the case of i, the case of j being similar (and easier). It suffices to check that for V ⊆ U an inclusion of rational subspaces of Spra(A, Gr A) with i −1 (U ) = i −1 (V ), we must have U = V . Let (A, Gr A) → (B, B Gr ) → (C, C Gr ) be the representing homomorphisms. By (b), B Gr = Gr B and C Gr = Gr C, so it suffices to check that B = C. However, for any ultrametric field E containing F , by [20, Lemma 2.2.9] we can check that B → C is an isomorphism by checking that B ⊗ F E → C ⊗ F E is an isomorphism. As in (b), we may thus reduce to the case where U, V are strictly rational subspaces; in this case we may appeal directly to [20,Corollary 2.5.13] to conclude.
The proofs of (d) and (e) are similar, so we omit the former. To prove (e), note that i is spectral and M(A) admits a basis of quasicompact open subsets for the special G-topology each of which is the inverse image of a quasicompact open subspace of Spra(A, Gr A). By (c), i also has dense image under the patch topology on Spra(A, Gr A). By Remark 1.12, we obtain a natural isomorphism Spec(D(M(A))) ∼ = Spra(A, Gr A), as desired.
Remark 9.7. The map j is injective when F has nontrivial norm (see Remark 6.12), but may not be injective when the norm on F is trivial (see Example 10.4).
Remark 9.8. The conclusion of Theorem 9.5(b) holds also for affinoid subdomains; see [20, Proposition 2.5.14(a)]. One may similarly extend Theorem 9.6(b) to affinoid subdomains; we omit futher details. Remark 9.9. As in Remark 2.9, the arguments found in [16] in the direction of Theorem 9.5 rely on elimination of quantifiers in the first-order theory of algebraically closed valued fields (ACVF); see especially the proof of [16,Theorem 4.1]. One can take a similar approach to Theorem 9.6 by establishing elimination of quantifiers in the theory of algebraically closed reified valued fields (which we propose to call ACRVF); this should follow easily from the corresponding result for ACVF since one is simply adding one constant to the language corresponding to the image of each positive real number in the value group. (A distinct but possibly related theory is the theory ACV 2 F of [15, §8].) On the other hand, it is also possible to deduce Theorem 9.6 directly from elimination of quantifers in ACVF, by making a base extension from F to a suitably large overfield as in the proof of Lemma 7.2; this approach is the one taken in [7].

Closed unit discs
We illustrate the previous discussion by making all of the constructions explicit in a simple but instructive case. The reader may find it useful to contrast this situation with the corresponding picture in the case of adic spectra [23, Example 2.20]. 5. A valuation of rank 2 which specializes a point of type 2. To describe these points more explicitly, choose x ∈ M(A) of type 2 for some particular z, ρ. The residue field k x of H(x) can then be identified with k(T ) with T being the class of (T − z)/λ for some λ ∈ K with |λ| = ρ. This defines an identification of k x with the function field of P 1 k , but this identification can be modified by changing the choices of z, λ; consequently, only the point at infinity on P 1 k is distinguished. What we can do canonically is to identify the finite places of k x with the branches of M(A) below x; each such place then defines a discrete valuation on k x , which we may compose with x to form a reified valuation of rank 2 specializing x. (For example, the branch of M(A) at x containing the type 1 point defined by the ideal (T −z) corresponds to a specialization of x in which the valuation of (T −z)/λ changes from being equal to 1 to being infinitesimally smaller than 1.) If x is not the Gauss point, then the infinite place of k x corresponds to the branch of M(A) above x, and we similarly obtain one more type 5 point specializing x. By contrast, if x is the Gauss point, one gets additional points of type 5 specializing x if and only if A + = A • ; see Example 10.3 for a typical example.
For any choice of A Gr whose r = 1 component equals A + /A •• , there is also a natural map Spa(A, A + ) → Spra(A, A Gr ). The complement of the image of this map consists of points of a sixth type. 6. A valuation of rank 2 which specializes a point of type 3. For x of type 3, there are exactly two points of type 6 specializing to x, corresponding to the branches of M(A) above and below x. To wit, if x is defined by some z, ρ, then T − z has valuation equal to ρ according to x, but infinitesimally larger or smaller than ρ according to the specializations. Under this identification, take A Gr = Gr K ⊂ Gr A. Let Γ be the reified value group of the valuation on K, and let Γ ′ be the lexicographic product Γ × R equipped with the reification inherited from Γ. Then Spra(A, A Gr ) contains a unique reified valuation v with values in Γ ′ extending the valuation on K and sending T to (0, 1); this is a type 5 point of Spra(A, A Gr ) not corresponding to a branch below the Gauss point and not satisfying v(T ) ≤ 1.
Example 10.4. Keep notation from Example 10.1, but now with the trivial norm on K. In this case, the structure of M(A) is simpler: the tree consists of branches corresponding to elements of K, meeting at the Gauss point. The lower endpoints of each branch is of type 1; the Gauss point is of type 2; other points are of type 3. There is again an embedding M(A) → Spra(A, A Gr ); the complement of its image consists of points of types 5 (specializing the Gauss point) and 6 (two for each type 3 point). However, one cannot fit Spa(A, A • ) in between; it arises from Spra(A, A Gr ) by removing the type 6 points, then collapsing the interior of each branch to a point.

Perfectoid algebras and their spectra
To conclude, we quickly redevelop the theory of perfectoid algebras in the context of reified adic spectra, following [20]. (See [23] and [11] for other treatments.) Throughout §11, fix a prime number p.
Definition 11.1. By a perfect uniform affinoid Banach algebra over F p , we will mean an affinoid seminormed ring (R ⊲ , R Gr ) such that R ⊲ is a perfect (i.e., the Frobenius map is bijective) uniform Banach algebra over F p (viewed as an ultrametric field using the trivial norm). Note that this forces R Gr to also be perfect. Note also that R ⊲ cannot be both perfect and noetherian unless it is a finite direct sum of perfect fields. Theorem 11.2. Let (R ⊲ , R Gr ) be a perfect uniform affinoid Banach algebra over F p . Let f : (R ⊲ , R Gr ) → (S ⊲ , S Gr ) be a morphism of affinoid seminormed rings satisfying one of the following conditions.
(a) The morphism f represents a rational subspace of Spra(S ⊲ , S Gr ). (b) The homomorphism R ⊲ → S ⊲ is finiteétale and S Gr is the integral closure of R Gr in Gr S ⊲ . Then (S ⊲ , S Gr ) is also a perfect uniform affinoid Banach algebra over F p . In particular, (R ⊲ , R Gr ) is really stably uniform, hence sheafy by Theorem 8. Definition 11.3. A uniform affinoid Banach algebra (A ⊲ , A Gr ) over Q p is perfectoid if for all r ∈ R + and x ∈ A +,r , there exists y ∈ A ⊲,•,r 1/p such that x − y p ∈ A +,r/p . Note that this forces y ∈ A +,r 1/p because A Gr is integrally closed.
Lemma 11.4. A uniform affinoid Banach algebra (A ⊲ , A Gr ) over Q p is perfectoid if and only if it satisfies the following conditions.
(b) There exists x ∈ A ⊲,• with x p − p ∈ p 2 A ⊲,• . In particular, the perfectoid condition depends only on A ⊲ and is consistent with the definition in [20].
Definition 11.5. Given a uniform affinoid Banach algebra (A ⊲ , A Gr ) over Q p , we may construct a perfect uniform affinoid Banach algebra (R ⊲ , R Gr ) over F p as follows. Define the underlying multiplicative monoid R ⊲ to be the inverse limit of A ⊲ under the p-power map. We define the addition on R ⊲ by the formula (x n ) n + (y n ) n = lim m→∞ (x m+n + y m+n ) p m n .
One checks easily that this gives R ⊲ the structure of a perfect uniform Banach algebra over F p with respect to the norm |(x n ) n | = |x n |. Similarly, define the underlying multiplicative monoid R Gr to be the inverse limit of A Gr under the p-power map; again, one checks easily that this gives (R ⊲ , R Gr ) the structure of a perfect uniform affinoid Banach algebra over F p . Definition 11.6. Let (R ⊲ , R Gr ) be a perfect uniform affinoid Banach algebra over F p . An element z = ∞ n=0 p n [z n ] ∈ W (R + ) is primitive of degree 1 if the following conditions hold: z 0 ∈ R ⊲,× ∩ R +,1/p , z −1 0 ∈ R +,p , z 1 ∈ (R + ) × . In this case, we can form a uniform affinoid Banach algebra (A ⊲ , A Gr ) over Q p by setting A ⊲ = W (R + )[[z] −1 ]/(z) and taking A Gr,r to be the image of R +,r under the composition of the Teichmüller map R ⊲ → W (R + )[[z] −1 ] with the projection to Gr A ⊲ .
Conversely, with notation as in Definition 11.5, the map θ : W (R + ) → A + induced by the multiplicative map R + → A + taking (x n ) n to x 0 is surjective and its kernel is principal with a generator which is primitive of degree 1 [20, Lemma 3.6.3].
Theorem 11.7. The constructions of Definitions 11.5 and 11.6 define quasiinverse functors which give equivalences of categories between the category of perfectoid uniform affinoid Banach algebras A over Q p and pairs (R, I) where R = (R ⊲ , R Gr ) is a perfect uniform affinoid Banach algebra over F p and I is an ideal of W (R + ) generated by an element which is primitive of degree 1.
Definition 11.8. Suppose that A and (R, I) correspond as in Theorem 11.7. Then A is an ultrametric field if and only if R is; consequently, in general we obtain a natural bijection Spra(A) → Spra(R).
Theorem 11.9. Suppose that A and (R, I) correspond as in Theorem 11.7.
(b) For U ⊆ Spra(A) and V ⊆ Spra(R) which correspond, U is a rational subspace if and only if V is. (c) With notation as in (b), let A → B and R → S be the morphisms representing U and V . Then B is a perfectoid uniform affinoid Banach algebra over Q p , S is a perfect uniform affinoid Banach algebra over F p , and B and (S, I · W (S + )) correspond as in Theorem 11.7.
Theorem 11.10. Suppose that A and (R, I) correspond as in Theorem 11.7.
(a) Let B ⊲ be a finiteétale A ⊲ -algebra viewed as a uniform Banach algebra (see [20,Proposition 2.8.16]) and let B Gr be the integral closure of A Gr in Gr A ⊲ . Then B = (B ⊲ , B Gr ) is again a perfectoid uniform affinoid Banach algebra over Q p . (b) The functors of Theorem 11.7 induce equivalences of categories of objects B as in (a) and pairs (S, IW (S + )) where R → S is a morphism as in Theorem 11.2(b) (so S ⊲ is a finiteétale R ⊲ -algebra).
Proof. This follows from [20, Theorem 3.6.21] and Lemma 11.4, without any further arguments required.
Theorem 11.11. Let A → B, A → C be morphisms of perfectoid uniform affinoid Banach algebras. Let (R, I) be the pair corresponding to A via Theorem 11.7, then apply the correspondence to A → B, A → C to obtain morphisms R → S, R → T . Then B ⊗ A C is again perfectoid and the map A → B ⊗ A C corresponds via Theorem 11.7 to the map R → S ⊗ R T ; moreover, the tensor product norm on B ⊗ A C induced by the spectral norms on B and C coincides with the spectral norm.
Proof. This is the analogue of [20, Proposition 3.6.11], but the proof of that statement is incomplete, so a corrected argument is needed. Note first that as in [20, Example 3.6.6], both claims hold in the case B = A{T 1 /ρ 1 , . . . , T n /ρ n }, C = A{T ′ 1 /ρ ′ 1 , . . . , T ′ n ′ /ρ ′ n ′ } with S = R{T 1 /ρ 1 , . . . , T n /ρ n }, T = R{T ′ 1 /ρ ′ 1 , . . . , T ′ n ′ /ρ ′ n ′ }. We may thus reduce the general case to the case where A → B, A → C factor through surjections B ′ → B, C ′ → C. Using [20, Remark 3.1.6, Lemma 3.3.9], we see that each of these surjections is almost optimal: the quotient norm induced by the spectral norm on the source coincides with the spectral norm on the target. We thus deduce both claims in the general case.
Definition 11.12. A reified perfectoid space is a reified adic space over Q p which is covered by the readic spectra of perfectoid affinoid Banach algebras over Q p . Using Theorem 11.7, Theorem 11.9, and Theorem 11.10, we may construct a "tilting" correspondence between reified perfectoid spaces and perfect uniform readic spaces over F p , which induces homeomorphisms of underlying topological spaces and ofétale topoi.