Adic tropicalizations and cofinality of Gubler models

Let $A\cong K\langle T_{1},\ldots,T_{n}\rangle/\mathfrak{a}$ be a strictly affinoid algebra. The Berkovich spectrum $\mathscr{M}(A)$ is the set of continuous seminorms $|-|\colon A\rightarrow\mathbb{R}_{\geqslant 0}$ that extend the given norm on $K$, equipped with the subspace topology for the inclusion $\mathscr{M}(A)\subset A^{\mathbb{R}_{\geqslant 0}}$. The Berkovich spectrum $\mathscr{M}(A)$ also carries a natural $G$-topology, which refines the ordinary topology, and can be described as follows.

Given elements $f_{1},\ldots,f_{m},g$ of $A$ with no common zero, let

be the corresponding rational algebra. The presentation $A\big{\langle}X_{1},\ldots,X_{m}\big{\rangle}\xtwoheadrightarrow{}B$ induces an inclusion $\mathscr{M}(B)\subset\mathscr{M}(A)$, and the subsets that occur in this way are called rational subdomains. A subset $V\subset\mathscr{M}(A)$ is admissible in the $G$-topology if every point $x\in V$ has a neighborhood of the form $V_{1}\cup\cdots\cup V_{n}$, where each $V_{i}$ is a rational domain, and $x\in V_{1}\cap\cdots\cap V_{n}$. Similarly, the admissible covers in the $G$-topology are covers of admissible subsets by admissible subsets $V=\bigcup_{i\in I}V_{i}$ such that every point $x\in V$ has a neighborhood which is a finite union of sets in the cover $V_{i_{1}}\cup\cdots\cup V_{i_{n}}$, with $x\in V_{i_{1}}\cap\cdots\cap V_{i_{n}}.$ The structure sheaf on $\mathscr{M}(A)$ is the unique sheaf of topological rings in this $G$-topology whose value on a rational domain $\mathscr{M}(B)$ is the rational algebra $B$, and whose restriction maps between these rational algebras are the natural ones. See [Ber90, Ber93] for further details.

Given a strictly affinoid algebra $A\cong K\langle T_{1},\ldots,T_{n}\rangle/\mathfrak{a}$, let $A^{\circ}\subset A$ be the subring of power bounded elements. The adic spectrum of the pair $(A,A^{\circ})$, denoted $\operatorname{Spa}(A,A^{\circ})$, is the set of all equivalence classes of continuous seminorms $\|-\|_{x}\colon A\longrightarrow\{0\}\sqcup\Gamma$ such that $\|a\|_{x}\leq 1$ for all $a\in A^{\circ}$.²²2The adic spectrum is defined similarly for pairs $(A,A^{+})$, where $A^{+}$ is a subring of the power bounded elements [Hub96, §1.1], but we will only need the case where $A^{+}=A^{\circ}$. Here $\Gamma$ denotes any totally ordered abelian group, written multiplicatively, and $\{0\}\sqcup\Gamma$ is the totally ordered abelian semigroup in which $0\cdot\gamma=0$ and $0<\gamma$ for all $\gamma\in\Gamma$, endowed with its order topology.

The equivalence relation is the smallest such that continuous seminorms

are equivalent whenever there is an inclusion $\alpha\colon\{0\}\sqcup\Gamma\hookrightarrow\{0\}\sqcup\Gamma^{\prime}$ of ordered abelian semigroups such that $\alpha{}_{{}^{\ \!\circ}}\|-\|_{x}=\|-\|_{x^{\prime}}$.

If $B=A\big{\langle}X_{1},\dots,X_{m}\big{\rangle}\big{/}(gX_{1}-f_{1},\ \!\dots,\ \!gX_{m}-f_{m})$ is a rational algebra, with the topology induced by the quotient norm, then the presentation $A\langle X_{1},\ldots X_{m}\rangle\xtwoheadrightarrow{}B$ induces an inclusion $\operatorname{Spa}(B,B^{\circ})\subset\operatorname{Spa}(A,A^{\circ})$. The subsets that occur in this way are called rational subsets, and rational subsets generate the topology on $\operatorname{Spa}(A,A^{\circ})$.

The structure sheaf $\mathscr{O}$ is the unique sheaf that takes the value $B$ on a rational subset $\operatorname{Spa}(B,B^{\circ})$, and whose restriction maps between such rational algebras are the natural ones. The stalk of this structure sheaf $\mathscr{O}$ at any point $x\in\operatorname{Spa}(A,A^{\circ})$ is a topological local ring, and the seminorm $\|-\|_{x}$ induces a continuous seminorm on $\mathscr{O}_{x}$. We also consider the subsheaf $\mathscr{O}^{\circ}$ of power bounded functions, whose value on an open subset $U$ is the ring of sections $f\in\mathscr{O}_{X}(U)$ for which $\|f\|_{x}\leq 1$ at every point $x\in U$. If $U=\operatorname{Spa}(B,B^{\circ})$ is a rational subset, then $\mathscr{O}^{\circ}(U)=B^{\circ}$.

A strictly $K$-analytic space is a Hausdorff topological space $X$ with a net of compact subspaces $V_{i}\subset X$, each equipped with a homeomorphism to a strictly $K$-affinoid space, and, for each inclusion $V_{j}\subset V_{i}$, a morphism of $K$-affinoid spaces identifying $V_{j}$ with an affinoid domain in $V_{i}$. We then consider $X$ as a locally topologically $G$-ringed space, with the induced $G$-topology.

An adic space over $K$ is a locally topologically ringed space $(Z,\mathscr{O}_{Z})$ with an atlas $\{V_{i}\!\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\!Z\}$, whose charts are adic spectra of strictly affinoid $K$-algebras.

There is a natural functor from $K$-analytic spaces to adic spaces, defined as follows. Suppose $V$ is a $K$-analytic space, with an atlas of Berkovich spectra $V_{i}\cong\mathscr{M}(A_{i})$, glued along inclusions induced by morphisms $f_{ij}\colon A_{i}\rightarrow A_{j}$ for $V_{j}\subset V_{i}$. Then the associated adic space $V^{\mathrm{ad}}$ has an atlas of adic spectra $\operatorname{Spa}(A_{i},A^{\circ}_{i})$, glued along the inclusions induced by the $\{f_{ij}\}$. If $X$ is a separated scheme of finite type over $K$, we write $X^{\mathrm{ad}}$ for the adic analytification of $X$, i.e., the adic space associated to the Berkovich analytification $X^{\mathrm{an}}$.

Many of the relations between adic spaces and $K$-analytic spaces, including those arising through analytification and tropicalization of algebraic schemes, are best understood in terms of Raynaud’s theory of admissible formal models. For an accessible presentation of this theory, see [Bos14].

A topological $R$-algebra $A$ is admissible if there is an isomorphism of topological rings

where $R\langle t_{1},\dots,t_{n}\rangle\big{/}\mathfrak{a}$ has its $\mathfrak{m}$-adic topology, such that:

• (i)

  the ideal $\mathfrak{a}\subset R\langle t_{1},\dots,t_{n}\rangle$ is finitely generated;

• (ii)

  the ring $R\langle t_{1},\dots,t_{n}\rangle\big{/}\mathfrak{a}$ is free of $\mathfrak{m}$-torsion.

An admissible formal $R$-scheme is a formal scheme over $\mathrm{Spf}_{\ \!\!}R$ that admits an open covering by formal spectra $\mathfrak{U}_{i}=\mathrm{Spf}A_{i}$, such that each $A_{i}$ is an admissible $R$-algebra. Throughout, we assume that all admissible formal schemes are separated and paracompact.

Let $\mathfrak{X}$ be an admissible formal $R$-scheme, and let $\big{\{}\mathfrak{U}_{i}\hookrightarrow\mathfrak{X}\big{\}}$ be an open cover of $\mathfrak{X}$ by formal spectra of admissible $R$-algebras $A_{i}$. Each topological algebra $K\otimes_{R}A_{i}$ is strictly $K$-affinoid [Bos14, §7.4]. Moreover, paracompactness of $\mathfrak{X}$ ensures that the Berkovich spectra $\mathscr{M}(K\otimes_{R}A_{i})$ of these strictly $K$-affinoid algebras glue to produce a $K$-analytic space over $K$ [Ber93, Proposition 1.3.3(b)]. We denote this $K$-analytic space $\mathfrak{X}^{\mathrm{an}}$; it is the Raynaud fiber of $\mathfrak{X}$.

Let $\mathfrak{X}$ be an admissible formal $R$-scheme. We write $\mathfrak{X}^{\mathrm{ad}}$ for the adic space associated to $\mathfrak{X}^{\mathrm{an}}$, and refer to this as the adic Raynaud fiber of $\mathfrak{X}$. It comes with a specialization morphism

of locally topologically ringed spaces over $\mathrm{Spec}_{\ \!}K$. Here, $\mathscr{O}^{\ \!\circ}_{\!\mathfrak{X}^{\mathrm{ad}}}\subset\mathscr{O}_{\!\mathfrak{X}^{\mathrm{ad}}}$ denotes the subsheaf of power bounded analytic functions on $\mathfrak{X}^{\mathrm{ad}}$.

When $\mathfrak{X}^{\mathrm{ad}}$ is quasi-compact, these specialization morphisms induce a natural isomorphism

in the category of locally topologically ringed spaces, where the inverse limit is taken over the category of formal models $\mathfrak{X}^{\prime}$ of $\mathfrak{X}^{\mathrm{an}}$ [Sch12, Theorem 2.22].

We briefly recall the basic notion of the ordinary extended tropicalization, hereafter referred to as tropicalization, for subschemes of toric varieties. See [Pay09, §3] for further details. Let $Y_{\Sigma}$ be the toric variety over $K$ associated to a fan $\Sigma$ in $N_{\mathbb{R}}$. Each cone $\sigma\in\Sigma$ corresponds to an affine torus-invariant open subvariety $U_{\sigma}\subset Y_{\Sigma}$, whose coordinate ring is the semigroup ring $K[S_{\sigma}]$ generated over $K$ by the semigroup $S_{\sigma}$ of characters of the dense torus that extend to regular functions on $U_{\sigma}$. We equip $\mathbb{R}\sqcup\{\infty\}$ with the topology that makes the exponential map $\mathbb{R}\sqcup\{\infty\}\longrightarrow\mathbb{R}_{\geq 0}$, given by $x\mapsto e^{-x}$, a homeomorphism. The tropicalization of $U_{\sigma}$ is the space of semigroup homomorphisms

with the topology induced by that of $\mathbb{R}\sqcup\{\infty\}$.

Just as $U_{\sigma}$ decomposes as a disjoint union of torus orbits $O_{\tau}$ corresponding to the faces $\tau\preceq\sigma$, the tropicalization $\text{Trop}(U_{\sigma})$ decomposes as a disjoint union of real vector spaces

Here each $N_{\mathbb{R}}/\text{span}(\tau)$ is canonically identified with the tropicalization $\operatorname{Trop}(O_{\tau})$ of the open torus orbit $O_{\tau}\subset U_{\sigma}$ corresponding to $\tau$. Each inclusion of faces $\tau\preceq\sigma$ induces open immersions $U_{\tau}\subset U_{\sigma}$ and $\operatorname{Trop}(U_{\tau})\subset\text{Trop}(U_{\sigma})$. The tropicalization $\operatorname{Trop}(Y_{\Sigma})$ is obtained by gluing $\{\text{Trop}(U_{\sigma})\}_{\sigma\in\Sigma}$ along the open immersions $\operatorname{Trop}(U_{\tau})\subset\text{Trop}(U_{\sigma})$, just as the toric variety $Y_{\Sigma}$ is obtained by gluing $\{U_{\sigma}\}_{\sigma\in\Sigma}$ along the open immersions $U_{\tau}\subset U_{\sigma}$. The natural tropicalization maps on affine open toric subvarieties glue to give a proper continuous surjection

Now, and for the remainder of the paper, we fix a separated scheme $X$ of finite type over $K$. Let $\iota\colon X\hookrightarrow Y_{\Sigma}$ be a closed embedding in a toric variety. Then the tropicalization

is the image of the closed subset $\iota(X)^{\mathrm{an}}\subset Y_{\Sigma}^{\mathrm{an}}$ under $\operatorname{Trop}$.

Let $\mathfrak{a}\subset K[M]$ denote the ideal cutting out $\iota(X)\cap T$, where $T$ denotes the dense torus in $Y_{\Sigma}$. Every point $v\in N_{\mathbb{R}}$ has an associated $R$-scheme

where

The special fiber of (1) is the initial degeneration of $X$ at $v$, denoted $\operatorname{in}_{v}(X);$ it is nonempty if and only if $v\in\mathrm{Trop}(X,\iota)$.

More generally, for any $w\in\operatorname{Trop}(Y_{\Sigma})$, there is a unique cone $\sigma\in\Sigma$ such that $w$ lies in the tropicalization $\operatorname{Trop}(O_{\sigma})\cong N_{\mathbb{R}}/\mathrm{span}(\sigma)$ of the open torus orbit $O_{\sigma}\subset Y_{\Sigma}$. In this case, $M_{\sigma}:=\text{span}(\sigma)^{\perp}\cap M$ is dual to the lattice $\text{span}(\sigma)\cap N$, and we define

The initial degeneration of $X$ at $w$, denoted $\operatorname{in}_{w}(\iota(X)\cap O_{\sigma})$, is the special fiber of the scheme $\text{Spec}_{\ \!}R[M_{\sigma}]^{w}\big{/}\big{(}\mathfrak{a}\cap R[M_{\sigma}]^{w}\big{)}$. For each cone $\sigma\in\Sigma$, the intersection of $\mathrm{Trop}(X,\iota)$ with $\operatorname{Trop}(O_{\sigma})$ is a finite polyhedral complex that parametrizes weight vectors on monomials in $M_{\sigma}$ such that $\operatorname{in}_{w}(\iota(X)\cap O_{\sigma})$ is nonempty. For simplicity, given $\iota\colon X\hookrightarrow Y_{\Sigma}$, $\sigma\in\Sigma$, and $w\in N_{\mathbb{R}}/\mathrm{span}(\sigma)$, we sometimes write $\operatorname{in}_{w}(X)$ for $\operatorname{in}_{w}(\iota(X)\cap O_{\sigma})$. With this notation,

Let $\Gamma\subset\mathbb{R}$ be the value group of $K$. As in the previous section, we consider the toric variety $Y_{\Sigma}$ over $K$ associated to a fan $\Sigma$ in $N_{\mathbb{R}}$.

Let $P\subset N_{\mathbb{R}}$ be a polyhedron, the intersection of finitely many closed halfspaces. Then the recession cone $\sigma_{P}$ is the closed polyhedral cone given by

Equivalently, the cone $\sigma_{P}$ is obtained by taking the closure of the cone over $P\times\{1\}$ in $N_{\mathbb{R}}\times\mathbb{R}$ and intersecting with $N_{\mathbb{R}}\times\{0\}$.

A polyhedron $P\subset N_{\mathbb{R}}$ is $(\Gamma,\Sigma)$-admissible if its recession cone $\sigma_{P}$ is in $\Sigma$, and $P$ itself can be expressed as an intersection of finitely many halfspaces

with $u_{i}$ in the character lattice $M$ and $\gamma_{i}$ in $\Gamma$. Note that any $(\Gamma,\Sigma)$-admissible polyhedron is pointed, i.e., its minimal faces have dimension zero, since the tail cone is in the fan $\Sigma$, which is a collection of pointed cones.

An extended $(\Gamma,\Sigma)$-admissible polyhedron is the closure $\overline{P}$ in $\operatorname{Trop}(Y_{\Sigma})$ of a pointed $(\Gamma,\Sigma)$-admissible polyhedron $P$ in $N_{\mathbb{R}}$. An extended $(\Gamma,\Sigma)$-admissible polyhedral complex $\Delta$ is a locally finite collection of extended $(\Gamma,\Sigma)$-admissible polyhedra, which we refer to as the faces of the complex, whose intersections with $N_{\mathbb{R}}$ form a polyhedral complex. The support $|\Delta|$ of $\Delta$ is the union of its faces, and $\Delta$ is complete if $|\Delta|=\operatorname{Trop}(Y_{\Sigma})$. Note that we require an extended $(\Gamma,\Sigma)$-admissible complex to be locally finite not only at points in the dense open subset $N_{\mathbb{R}}\cap|\Delta|$, but at every point $w\in|\Delta|$. See also Remark 3.3.1, below.

We will use the following lemma, on existence of complete complexes that simultaneously refine any finite collection of admissible polyhedra, in the proofs of Theorems  1.2 and 4.1.3. It is an immediate consequence of the following result of Coles and Friedenburg.

If $\overline{P}\subset\operatorname{Trop}(Y_{\Sigma})$ is an extended $(\Gamma,\Sigma)$-admissible polyhedron, then $\operatorname{Trop}^{-1}(\overline{P})$ is a strictly affinoid analytic domain in $Y_{\Sigma}^{\mathrm{an}}$, and is canonically realized as the Raynaud fiber of the formal completion of a flat $R$-scheme whose generic fiber is the affine open subvariety $U_{\sigma_{P}}\subset Y_{\Sigma}$, as we now explain. Let $R[M]^{P}$ denote the $R$-algebra

and let $\mathscr{U}_{P}$ denote the $R$-scheme $\mathscr{U}_{P}=\mathrm{Spec}\ \!R[M]^{P}$. By [Gub13, Propositions 6.6 and 6.10], $\mathscr{U}_{P}$ is a normal scheme flat over $R$ with generic fiber $(\mathscr{U}_{P})_{K}:=\mathscr{U}_{P}\otimes_{R}K$ naturally isomorphic to the affine toric variety associated to the recession cone $\sigma_{P}$:

Let $\mathfrak{U}_{P}$ denote the formal $R$-scheme obtained as the completion of $\mathscr{U}_{P}$ along its special fiber. Since $K$ is algebraically closed, the value group $\Gamma$ is divisible, and hence the formal scheme $\mathfrak{U}_{P}$ is admissible [Gub13, Proposition 6.7].

Let $\mathfrak{U}_{P}^{\mathrm{an}}$ denote the Raynaud fiber of $\mathfrak{U}_{P}$. Then the Raynaud fiber $\mathfrak{U}_{P}^{\mathrm{an}}$ is naturally identified with a strictly affinoid domain inside the Berkovich analytification $U_{\sigma_{P}}^{\mathrm{an}}$ [Gub13, §4.13]. Specifically,

Applying [Gub13, Lemma 6.21] to each orbit $O_{\sigma}\subset Y_{\Sigma}$, for $\sigma\in\Sigma$, we see that this strictly affinoid domain is the inverse image of $\overline{P}\subset\operatorname{Trop}(Y_{\Sigma})$ under the tropicalization map,

These models of strictly affinoid domains associated to extended $(\Gamma,\Sigma)$-admissible polyhedra glue together naturally, as follows. Each inclusion of a face of an extended $(\Gamma,\Sigma)$-admissible polyhedron $\overline{Q}\subset\overline{P}$ induces a Zariski open embedding of $R$-schemes $\mathscr{U}_{Q}\subset\mathscr{U}_{P}$ and a Zariski open embedding of formal $R$-schemes $\mathfrak{U}_{Q}\subset\mathfrak{U}_{P}$. If $\Delta$ is an extended $(\Gamma,\Sigma)$-admissible polyhedral complex, gluing along these open embeddings naturally produces a flat $R$-scheme $\mathscr{Y}_{\Delta}$ and an admissible formal $R$-scheme $\mathfrak{Y}_{\Delta}$, respectively. By construction, $\mathfrak{Y}_{\Delta}$ is the formal completion of $\mathscr{Y}_{\Delta}$ along its special fiber. Furthermore, the Raynaud fiber $\mathfrak{Y}_{\Delta}^{\mathrm{an}}$ is the preimage of the support $|\Delta|\subset\operatorname{Trop}(Y_{\Sigma})$ under the tropicalization map, i.e.,

Let $\iota\colon X\hookrightarrow Y_{\Sigma}$ be the inclusion of a closed subscheme, and let $\Delta$ be an extended $(\Gamma,\Sigma)$-admissible polyhedral complex in $\operatorname{Trop}(Y_{\Sigma})$. Then, for each face $\overline{P}$ of $\Delta$, we obtain an $R$-model $\mathscr{X}_{P}$ of $X\cap U_{\sigma_{P}}$ as the closure of $X$ in the toric scheme $\mathscr{U}_{P}$. Gluing along the inclusions of faces in $\Delta$, we obtain an $R$-model $\mathscr{X}_{\Delta}$ of $X$, which we refer to as the (algebraic) Gubler model of $X$ associated to the pair $(\iota,\Delta)$. See also [CF23a] for further discussion of such formal models.

Completing each $R$-scheme $\mathscr{X}_{P}$ along its special fiber, we obtain an admissible formal $R$-scheme $\mathfrak{X}_{P}$. Gluing these formal schemes along the inclusions in $\Delta$, we obtain an admissible formal $R$-scheme $\mathfrak{X}_{\Delta}$ isomorphic to the completion of $\mathscr{X}_{\Delta}$ along its special fiber, whose Raynaud fiber $\mathfrak{X}_{\Delta}^{\mathrm{an}}$ is the preimage of $|\Delta|$ in $X^{\mathrm{an}}$. In particular, $\mathfrak{X}_{\Delta}$ is a formal model of $X^{\mathrm{an}}$ if and only if $|\Delta|$ contains $\mathrm{Trop}(X,\iota)$. In this case, we say that $\Delta$ covers $\mathrm{Trop}(X,\iota)$ and refer to $\mathfrak{X}_{\Delta}$ as the formal Gubler model of $X^{\mathrm{an}}$ associated to the pair $(\iota,\Delta)$. When we want to stress the role of the closed embedding $\iota\colon X\hookrightarrow Y_{\Sigma}$ in the construction of the formal Gubler model $\mathfrak{X}_{\Delta}$, we write $\mathfrak{X}_{(\iota,\Delta)}$.

Let $\Delta$ and $\Delta^{\prime}$ be extended $(\Gamma,\Sigma)$-admissible polyhedral complexes that cover $\mathrm{Trop}(X,\iota)$. We say that $\Delta^{\prime}$ refines $\Delta$ if each face $\overline{P}^{\prime}$ of $\Delta^{\prime}$ is contained in some face $\overline{P}$ of $\Delta$. The induced maps $\mathfrak{X}_{P^{\prime}}\rightarrow\mathfrak{X}_{P}$ glue to produce a morphism of formal $R$-schemes $\mathfrak{X}_{\Delta^{\prime}}\longrightarrow\mathfrak{X}_{\Delta}$. This gives a functor from the category of extended $(\Gamma,\Sigma)$-admissible complexes that cover $\mathrm{Trop}(X,\iota)$, in which the morphisms are refinements, to the category of admissible formal models of $X^{\mathrm{an}}$.

Consider the category of toric embeddings of $X$, whose objects are closed embeddings $\iota\colon X\hookrightarrow Y_{\Sigma}$ into toric varieties, and whose morphisms are commutative diagrams

induced by a toric morphism $\phi\colon\Sigma\longrightarrow\Sigma^{\prime}$ of fans. Given such a diagram, let $\Delta^{\prime}$ be an extended $(\Gamma,\Sigma)$-admissible polyhedral complex that covers $\mathrm{Trop}(X,\iota^{\prime})$. Note that we can refine any extended $(\Gamma,\Sigma)$-admissible polyhedral complex $\Delta$ that covers $\mathrm{Trop}(X,\iota)$ to obtain a cover $\Delta^{+}$ such that the induced morphism $\operatorname{Trop}(f)\colon\operatorname{Trop}(Y_{\Sigma})\longrightarrow\operatorname{Trop}(Y_{\Sigma^{\prime}})$ maps each $\overline{P}\in\Delta^{+}$ into some $\overline{P}^{\prime}\in\Delta^{\prime}$. There are then induced morphisms of algebraic and admissible formal $R$-schemes

which glue to give morphisms of algebraic and formal Gubler models

From this it follows that each morphism $f$ of toric embeddings induces a morphism of adic tropicalizations,

in the category of locally topologically ringed spaces; hence, adic tropicalization is a functor from toric embeddings to locally topologically ringed spaces.

We now show that the disjoint union of the initial degenerations $\operatorname{in}_{w}(X)$, for $w\in\operatorname{Trop}(X,\iota)$, is naturally identified with the underlying set of the adic tropicalization $\mathfrak{Trop}(X,\iota)$.

As in the previous sections, we let $X$ denote a separated scheme of finite type over $K$. Recall that a formal scheme $\mathfrak{X}$ is algebraizable if it is isomorphic to the formal completion of a flat $R$-scheme of finite type. By construction, any formal Gubler model is algebraizable.

We now state and prove a technical theorem about the Gubler models $\mathfrak{X}_{\Delta}$ of projective schemes.