2. Skeletons, formal models and divisors [0598]

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2. Skeletons, formal models and divisors

In this section we first define formal schemes and their generic and special fibres. For details we refer to [Bos14, II.7, II.8.3]. Then we recall the concept of skeletons associated to strongly nondegenerate strictly polystable formal schemes introduced by Berkovich in [Ber99]. To a subdivision of the skeleton, one can associate a formal analytic structure as in [Gub10, Proposition 5.5]. We generalize the subsequent results of [Gub10, §5] concerning the stratum face correspondence by dropping the condition of algebraically closedness of the base field. Finally we explain how a piecewise affine linear function on the skeleton induces a Cartier divisor on the formal scheme corresponding to a suitable subdivision of the skeleton.

Definition 2.1.

Let $Y$ be a reduced scheme of locally finite type over a field $\kappa$ . Set $Y^{(0)}:=Y$ and let $Y^{(i+1)}$ be the complement of the set of normal points in $Y^{(i)}$ . The irreducible components of $Y^{(i)}\setminus Y^{(i+1)}$ are called strata of $Y$ . There is a partial ordering on the set of strata given by $R_{1}\leq R_{2}$ if and only if $\overline{R_{1}}\subseteq\overline{R_{2}}$ . A cycle $Z\in Z(Y)$ is called a strata cycle if there are strata $S_{1},...,S_{n}$ of $Y$ such that $Z=\sum m_{i}\overline{S_{i}}$ with $m_{i}\in\mathbb{R}$ .

Definition 2.2.

A topological ring $A$ is called adic if there is an ideal $\mathfrak{a}\subseteq A$ such that the ideals $(\mathfrak{a}^{n})_{n\in\mathbb{N}}$ form a neighbourhood basis for $0$ . We call $\mathfrak{a}$ a defining ideal. Let $A$ be an adic, complete, separated ring with finitely generated defining ideal $\mathfrak{a}$ . The affine formal scheme of $A$ is the locally topologically ringed space $\Spf(A)=(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ where $\mathfrak{X}$ and $\mathcal{O}_{\mathfrak{X}}$ are defined as follows: $\mathfrak{X}$ is the set of all open prime ideals of $A$ . As a prime ideal is open if and only if it contains $\mathfrak{a}$ , we may identify $\mathfrak{X}$ with $\Spec(A/\mathfrak{a})\subseteq\Spec(A)$ and we endow $\mathfrak{X}$ with the topology induced by the Zariski topology on $\Spec(A)$ . Moreover we define

\mathcal{O}_{\mathfrak{X}}:=\underset{\leftarrow}{\lim}\;\mathcal{O}_{\Spec(A/\mathfrak{a}^{n})}.

A formal scheme is a locally topologically ringed space $(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ such that for each $x\in\mathfrak{X}$ there is an open neighbourhood $\mathfrak{U}$ of $x$ with $\left(\mathfrak{U},\mathcal{O}_{\mathfrak{X}}\Big|_{\mathfrak{U}}\right)$ isomorphic to an affine formal scheme.

Now let $\mathfrak{a}$ be a defining ideal of $K^{\circ}$ . A topological $K^{\circ}$ -algebra $A$ is called admissible, if $\left\{a\in A\;\Big|\;\mathfrak{a}^{n}\cdot a=0\text{ for some }n\in\mathbb{N}\right\}=\{0\}$ i.e. $A$ does not have $K^{\circ}$ -torsion and if $A$ is isomorphic to a $K^{\circ}$ -algebra of the form $K^{\circ}\langle\zeta_{1},...,\zeta_{n}\rangle/(a_{1},...,a_{m})$ endowed with the $\mathfrak{a}$ -adic topology. A formal $K^{\circ}$ -scheme $\mathfrak{X}$ is called admissible if there is a locally finite open cover $(\mathfrak{U}_{i})_{i\in I}$ of $\mathfrak{X}$ with $\mathfrak{U}_{i}=\Spf(A_{i})$ for admissible $K^{\circ}$ -algebras $A_{i}$ .

Let $\mathfrak{X}=\Spf(A)$ be an admissible formal affine $K^{\circ}$ -scheme. The analytic generic fibre of $\mathfrak{X}$ is defined as $\mathfrak{X}^{\textup{an}}:=\mathcal{M}(A\otimes_{K^{\circ}}K)$ , where $\mathcal{M}(\cdot)$ denotes the Berkovich spectrum (cf. [Ber90, 1.2]). The special fibre of $\mathfrak{X}$ is given by $\tilde{\mathfrak{X}}:=\Spec(A\otimes_{K^{\circ}}k)$ , where $k:=K^{\circ}/K^{\circ\circ}$ is the residue field of $K$ . For an admissible formal $K^{\circ}$ -scheme $\mathfrak{X}$ one obtains the generic and the special fibre by a gluing process. There is a canonical surjective reduction map $\red:\mathfrak{X}^{\textup{an}}\rightarrow\tilde{\mathfrak{X}}$ , see [GRW17, §2.13].

Definition 2.3.

For $n\in\mathbb{N}_{>0}$ and $a\in K^{\circ\circ}$ we define

\mathfrak{X}(n,a):=\Spf(K^{\circ}\langle x_{0},...,x_{n}\rangle/(x_{0}...x_{n}-a)).

For tuples $\boldsymbol{n}=(n_{0},...,n_{p})\in\mathbb{N}_{>0}^{p+1}$ and $\boldsymbol{a}=(a_{0},...,a_{p})\in(K^{\circ\circ})^{p+1}$ we define $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a}):=\mathfrak{X}(n_{0},a_{0})\times_{K^{\circ}}...\times_{K^{\circ}}\mathfrak{X}(n_{p},a_{p})$ and for $m\in\mathbb{N}$ we set $\mathfrak{X}(m):=\mathfrak{X}(m,1)$ . A strictly polystable formal scheme over $K^{\circ}$ is an admissible formal scheme $\mathfrak{X}$ over $K^{\circ}$ which can be covered by formal open sets $\mathfrak{U}$ with étale morphisms

\psi:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m):=\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})\times_{K^{\circ}}\mathfrak{X}(m)

where $\boldsymbol{n}$ , $\boldsymbol{a}$ and $m$ may depend on $\mathfrak{U}$ . We say that $\mathfrak{X}$ is strongly nondegenerate strictly polystable if all $a_{i}$ can be chosen nonzero.

To a strongly nondegenerate strictly polystable formal scheme $\mathfrak{X}$ over $K^{\circ}$ Berkovich introduced in [Ber99] a canonical polytopal subset $S(\mathfrak{X})$ of $\mathfrak{X}^{\textup{an}}$ called the skeleton. It is a closed subset of $\mathfrak{X}^{\textup{an}}$ which is locally given by canonical polysimplices and can be described as follows. Let $\psi:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ be an étale morphism as above. The generic fibre of the right hand side is given as $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)^{\textup{an}}=\mathscr{M}(A)$ where $A=(K\langle T_{0}^{\pm},...,T_{m}^{\pm}\rangle)\langle T_{00},...,T_{p,n_{p}}\rangle/(T_{00}...T_{0,n_{0}}-a_{0},...,T_{p0}...T_{p,n_{p}}-a_{p})$ . The elements of $A$ can be expressed as $\sum_{\mu}a_{\mu}T^{\mu}$ with $a_{\mu}\in K\langle T_{0}^{\pm},...,T_{m}^{\pm}\rangle$ and $a_{\mu}=0$ if there is an $i\in\{0,...,p\}$ such that $\mu_{i,k}\geq 1$ for all $k\in\{0,...,n_{i}\}$ . Now to an element $\boldsymbol{t}$ in the polysimplex $\left\{\boldsymbol{t}\in\mathbb{R}_{\geq 0}^{\boldsymbol{n}+\boldsymbol{1}}\;\Big|\;t_{i0}+...+t_{in_{i}}=-\log(|a_{i}|),0\leq i\leq p\right\}$ we associate a seminorm on $A$ by sending a power series as above to $\max_{\mu}\{|a_{\mu}|\exp(-\boldsymbol{t}\cdot\mu)\}$ . This gives an embedding of the polysimplex into $\mathscr{M}(A)$ whose image is denoted by $\Delta$ . The skeleton $S(\mathfrak{U})$ of $\mathfrak{U}$ is defined to be $(\psi^{\textup{an}})^{-1}(\Delta)$ . One can show that $\psi^{\textup{an}}$ induces a homeomorphism from $(\psi^{\textup{an}})^{-1}(\Delta)$ to $\Delta$ if $\mathfrak{U}$ has a unique minimal stratum which maps to the minimal stratum of $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ . The skeleton $S(\mathfrak{X})$ of $\mathfrak{X}$ is the union of all $S(\mathfrak{U})$ and is independent of all choices.

To a stratum $S$ of $\mathfrak{X}$ one can associate a canonical polysimplex $\Delta_{S}$ in the skeleton such that the interiors of the $\Delta_{T}$ form a disjoint cover of $S(\mathfrak{X})$ where $T$ ranges over all strata of $\tilde{\mathfrak{X}}$ . In order to do so, we choose a refinement of the cover of $\mathfrak{X}$ as described in the Proposition below and choose $\mathfrak{U}$ such that $S$ is its distinguished stratum. We then define $\Delta_{S}:=S(\mathfrak{U})$ .

An admissible formal scheme $\mathfrak{X}$ is called strongly nondegenerate polystable if there exists a strongly nondegenerate strictly polystable formal scheme $\mathfrak{X}^{\prime}$ and a surjective étale morphism $\mathfrak{X}^{\prime}\rightarrow\mathfrak{X}$ . The skeleton of $\mathfrak{X}$ is defined to be the image of the skeleton of $\mathfrak{X}^{\prime}$ under the map $\mathfrak{X}^{\prime\textup{an}}\rightarrow\mathfrak{X}^{\textup{an}}$ .

One can endow the skeleton with a piecewise linear structure, see [Ber04, §6]. We will define piecewise affine linear functions on the skeleton of a strongly nondegenerate strictly polystable formal scheme in Definition 2.10. There is a canonical continuous retraction map $p_{\mathfrak{X}}:\mathfrak{X}^{\textup{an}}\rightarrow S(\mathfrak{X})$ which restricts to the identity on $S(\mathfrak{X})$ . For details see [Ber99, §4], [Ber04, §4] or [Gub10, 5.3].

We have the following stratum face correspondence due to Berkovich:

Proposition 2.4.

Let $\mathfrak{X}$ be a strongly nondegenerate polystable formal scheme with skeleton $\Delta$ . There is a bijective correspondence between the open faces of $\Delta$ and the strata of $\tilde{\mathfrak{X}}$ given by

R=\red(p_{\mathfrak{X}}^{-1}(\tau)),\hskip 56.9055pt\tau=p_{\mathfrak{X}}(\red^{-1}(R)).

Proof.

[Ber99, Theorem 5.2 (iv), Theorem 5.4]. ∎

Proposition 2.5.

Let $\mathfrak{X}$ be a strongly nondegenerate strictly polystable formal scheme over $K^{\circ}$ . Any formal open covering of $\mathfrak{X}$ admits a refinement $\{\mathfrak{U}^{\prime}\}$ by formal open subsets $\mathfrak{U}^{\prime}$ as in Definition 2.3 such that

i)

Every $\mathfrak{U}^{\prime}$ is a formal affine open subscheme of $\mathfrak{X}$ ,
ii)

there is a distinguished stratum $S$ of $\tilde{\mathfrak{X}}$ associated to $\mathfrak{U}^{\prime}$ such that for any stratum $T$ of $\tilde{\mathfrak{X}}$ , we have $S\subseteq\overline{T}$ if and only if $\tilde{\mathfrak{U}^{\prime}}\cap\overline{T}\neq\emptyset$ ,
iii)

$\tilde{\psi}^{-1}(\{\tilde{\boldsymbol{0}}\}\times\widetilde{\mathfrak{X}(m)})$ is the stratum of $\tilde{\mathfrak{U}^{\prime}}$ which is equal to $\tilde{\mathfrak{U}^{\prime}}\cap S$ for the distinguished stratum $S$ associated to $\mathfrak{U}^{\prime}$ ,
iv)

every stratum of $\tilde{\mathfrak{X}}$ is the distinguished stratum of a suitable $\mathfrak{U}^{\prime}$ .

Proof.

The very same arguments as in [Gub10, Proposition 5.2] apply to our situation. ∎

From now on let $\mathfrak{X}$ be a strongly nondegenerate strictly polystable formal scheme over $K^{\circ}$ and denote by $\Gamma$ the value group of $K$ . For the basic notions of convex geometry we refer to [Gub13, Appendix A]. We will work with $\Gamma$ -rational polytopal subdivisions $\mathfrak{D}$ of $S(\mathfrak{X})$ , i.e. $\mathfrak{D}$ is a family of $\Gamma$ -rational polytopes contained in a canonical polysimplex such that for every stratum $S$ of $\tilde{\mathfrak{X}}$ the set $\left\{\Delta\in\mathfrak{D}\;\Big|\;\Delta\subseteq\Delta_{S}\right\}$ is a polytopal decomposition of $\Delta_{S}$ . Here a polytopal decomposition means a finite family of polytopes covering $\Delta_{S}$ which is closed under taking faces and such that the intersection of two polytopes in the family is a face of both and a $\Gamma$ -rational polytope means a polytope which is defined by inequalities of the form $\boldsymbol{m}\boldsymbol{x}+c\geq 0$ with $\boldsymbol{m}\in\mathbb{Z}^{r},c\in\Gamma$ .

Construction 2.6.

Let $\mathfrak{D}$ be such a subdivision. We will construct a canonical formal scheme $\mathfrak{X}^{\prime\prime}$ over $K^{\circ}$ associated to $\mathfrak{D}$ together with a morphism $\iota:\mathfrak{X}^{\prime\prime}\rightarrow\mathfrak{X}$ which induces the identity on the generic fibre such that there is a one to one correspondence between the open faces of $\mathfrak{D}$ and the strata of $\tilde{\mathfrak{X}}^{\prime\prime}$ . First of all we choose a covering of $\mathfrak{X}$ as in Proposition 2.5. Let $\mathfrak{U}$ be a member of this covering with an étale morphism $\psi:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ and let $S$ be the distinguished stratum of $\mathfrak{U}$ . For $\Delta\in\mathfrak{D}\cap\Delta_{S}$ we set

A^{\prime}:=\left\{\sum_{\mu}a_{\mu}T^{\mu}\in K((T_{00},...,T_{p,n_{p}}))\;\Big|\;\forall_{u\in\Delta}:\;\lim v(a_{\mu})+u\cdot\mu=\infty\right\}

and $A:=A^{\prime}/(T_{00}...T_{0,n_{0}}-a_{0},...,T_{p0}...T_{p,n_{p}}-a_{p})$ and define

A^{\Delta}:=\left\{\sum_{\mu}a_{\mu}T^{\mu}\in A\;\Big|\;\forall_{u\in\Delta,\mu\in\mathbb{Z}^{\boldsymbol{n}+\boldsymbol{1}}}:\;v(a_{\mu})+\mu\cdot u\geq 0\right\}

and $\mathfrak{U}_{\Delta}:=\Spf A^{\Delta}$ . If $\Delta_{1},\Delta_{2}\in\mathfrak{D}\cap\Delta_{S}$ then $\Delta_{1}\cap\Delta_{2}$ is a face of both and by transferring the arguments in [Gub13, Proposition 6.12] to the analytic situation, we obtain that the canonical morphisms $\mathfrak{U}_{\Delta_{1}\cap\Delta_{2}}\rightarrow\mathfrak{U}_{\Delta_{i}}$ are open immersions. Hence we can glue the $\mathfrak{U}_{\Delta}$ along this data to obtain a formal scheme which we denote by $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}$ together with a morphism $\iota^{\prime}:\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})$ . Let $\psi^{\prime}:\mathfrak{U}^{\prime\prime}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}\times\mathfrak{X}(m)$ be the base change of $\psi$ with respect to $\iota^{\prime}\times\Id$ . The construction of $\mathfrak{U}^{\prime\prime}$ does not depend on the choice of $\psi$ up to isomorphism: Let $\rho:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ be another étale morphism. Then up to reordering the coordinates, $\rho^{\ast}x_{i}=u_{i}\psi^{\ast}x_{i}$ for some $u_{i}\in\mathcal{O}(\mathfrak{U})^{\times}$ . Then we have canonical $K^{\circ}$ -algebra isomorphisms:

	$\displaystyle\mathcal{O}(\mathfrak{U})\hat{\otimes}_{\psi^{\ast}}A^{\Delta}$	$\displaystyle\rightarrow\mathcal{O}(\mathfrak{U})\hat{\otimes}_{\rho^{\ast}}A^{\Delta},$
	$\displaystyle a\otimes x_{i}$	$\displaystyle\mapsto u_{i}a\otimes x_{i},$

which yield an isomorphism of the $\mathfrak{U}^{\prime\prime}$ constructed with $\psi$ respectively $\rho$ .
We glue the $\mathfrak{U}^{\prime\prime}$ to obtain our formal scheme $\mathfrak{X}^{\prime\prime}$ . Although $\mathfrak{X}^{\prime\prime}$ might not be admissible, we can define its generic fibre and reduction map in the usual way as the algebras $A^{\Delta}\otimes_{K^{\circ}}K$ are strictly $K$ -affinoid (see [Gub13, Proposition 6.17]). Then $\iota$ induces the identity on the generic fibres and we set $p_{\mathfrak{X}^{\prime\prime}}:=p_{\mathfrak{X}}$ . Note that $\mathfrak{X}^{\prime\prime}$ is admissible if the vertices of the polytopes in $\mathfrak{D}$ are $\Gamma$ -rational, in particular the base change of $\mathfrak{X}^{\prime\prime}$ to the valuation ring of the completion of an algebraic closure of $K$ is admissible, see [Gub13, Proposition 6.7].

Remark 2.7.

If $\mathfrak{D}$ is trivial i.e. $\Delta\in\mathfrak{D}$ only if $\Delta=\Delta_{S}$ for some stratum $S$ of $\tilde{\mathfrak{X}}$ then it is an immediate consequence from the construction that $\mathfrak{X}^{\prime\prime}=\mathfrak{X}$ .

We will frequently use the following generalization of [Gub10, Proposition 5.7] which is a stratum face correspondence for the $\mathfrak{X}^{\prime\prime}$ constructed above.

Proposition 2.8.

Let $\mathfrak{X}$ be a strongly nondegenerate strictly polystable formal scheme with skeleton $\Delta$ and $\mathfrak{D}$ a subdivision of $\Delta$ with associated formal structure $\mathfrak{X}^{\prime\prime}$ . Then there is a bijective correspondence between the open faces of $\mathfrak{D}$ and the strata of $\tilde{\mathfrak{X}}^{\prime\prime}$ given by

R=\red(p_{\mathfrak{X}^{\prime\prime}}^{-1}(\tau)),\hskip 56.9055pt\tau=p_{\mathfrak{X}^{\prime\prime}}(\red^{-1}(R)).

Furthermore, in the second equality, $R$ can be replaced by any nonempty subset of $R$ .

Proof.

We follow the proof of [Gub10, Proposition 5.7] but in order to establish the result for an arbitrary non-archimedean field $K$ (not necessarily algebraically closed), we use [Gub13, Proposition 6.22] instead of [Gub07, Proposition 4.4]. Let $\tau$ be an open face of $\mathfrak{D}$ . We prove first that $R:=\red(p_{\mathfrak{X}^{\prime\prime}}^{-1}(\tau))$ is a stratum of $\tilde{\mathfrak{X}}^{\prime\prime}$ . There is a unique stratum $S$ of $\tilde{\mathfrak{X}}$ such that $\tau$ is contained in the interior of $\Delta_{S}$ . Let $\mathfrak{U}$ be a formal open subset of $\mathfrak{X}$ such that $S$ is the distinguished stratum of $\mathfrak{U}$ (Proposition 2.5). As strata are compatible with localization we may assume $\mathfrak{X}=\mathfrak{U}$ . Let $\psi_{1}^{\prime}:\mathfrak{X}^{\prime\prime}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}$ be the base change of the composition of the étale map $\psi:\mathfrak{X}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ with the projection on the first factor $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})$ . By [Gub13, Proposition 6.22] the first part of the proposition holds for $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}$ . Let $T$ be the stratum of $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}$ corresponding to $\tau$ , i.e.

(2.1)

\displaystyle\tau=p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}(\red^{-1}(T))

and

(2.2)

\displaystyle T=\red(p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}^{-1}(\tau)).

where $p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}:\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime an}\rightarrow\Delta_{S}$ is the retraction map. We prove $R=\tilde{\psi}_{1}^{\prime-1}(T)$ . First we observe that

\red(({\psi^{\prime}}_{1}^{\textup{an}})^{-1}(p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}^{-1}(\tau)))=\tilde{\psi}_{1}^{\prime-1}(\red(p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}^{-1}(\tau))).

The inclusion $\subseteq$ is clear because $\red\circ{\psi^{\prime}}_{1}^{\textup{an}}=\tilde{\psi}^{\prime}_{1}\circ\red$ . The other inclusion follows from this fact and an application of [Gub13, Proposition 6.22]. For details we refer to the proof of [Gub10, Proposition 5.7]. We conclude

R=\red(p_{\mathfrak{X}^{\prime\prime}}^{-1}(\tau))=\red(({\psi^{\prime}}_{1}^{\textup{an}})^{-1}(p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}^{-1}(\tau)))=\tilde{\psi}_{1}^{\prime-1}(\red(p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}^{-1}(\tau)))\overset{(\ref{T=})}{=}\tilde{\psi}_{1}^{\prime-1}(T).

By [Ber99, Lemma 2.2] $R$ is a strata subset. To see that $R$ is indeed a stratum it is enough to show that $R$ is irreducible. But this follows from

\tilde{\psi}_{1}^{\prime-1}(T)=(T\times\tilde{\mathfrak{X}}(m))\times_{\tilde{\mathfrak{X}}(\boldsymbol{n},\boldsymbol{a},m)^{\prime}}\tilde{\mathfrak{X}}^{\prime\prime}\cong(T\times\tilde{\mathfrak{X}}(m))\times_{\{\tilde{0}\}\times\tilde{\mathfrak{X}}(m)}\tilde{\psi}^{-1}(\{\tilde{0}\}\times\tilde{\mathfrak{X}}(m))\cong T\times S,

where the latter is irreducible by [Gro65, Corollaire 4.5.8 (i)]. As the open faces of $\mathfrak{D}$ cover $\Delta$ , every stratum of $\tilde{\mathfrak{X}}^{\prime\prime}$ is obtained this way. It remains to prove that we can recover $\tau$ from $R$ . First note that

p_{\mathfrak{X}^{\prime\prime}}(({\psi^{\prime}}_{1}^{\textup{an}})^{-1}(\red^{-1}(T)))=p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}(\red^{-1}(T)).

The inclusion $\subseteq$ is clear because $p_{\mathfrak{X}^{\prime\prime}}=p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}\circ{\psi^{\prime}}_{1}^{\textup{an}}$ . For the other inclusion, let $x\in p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}(\red^{-1}(T))=\tau$ . As the sets $\red^{-1}(T^{\prime})$ with $T^{\prime}$ varying over the strata of $\tilde{\mathfrak{X}}(\boldsymbol{n},\boldsymbol{a})^{\prime}$ cover ${\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}^{\textup{an}}$ and using [Gub13, Proposition 6.22] and the fact the $p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}$ restricts to the identity on $\Delta$ we deduce $x\in\red^{-1}(T)$ . Hence $x$ is an element of the left hand side which proves the equality claimed in the display. Now the rest is an easy calculation:

	$\displaystyle p_{\mathfrak{X}^{\prime\prime}}(\red^{-1}(R))$	$\displaystyle=p_{\mathfrak{X}^{\prime\prime}}(\red^{-1}(\tilde{\psi}_{1}^{\prime-1}(T)))$
		$\displaystyle=p_{\mathfrak{X}^{\prime\prime}}(({\psi^{\prime}}_{1}^{\textup{an}})^{-1}(\red^{-1}(T)))$
		$\displaystyle=p_{\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\prime}}(\red^{-1}(T))$
		$\displaystyle\overset{(\ref{tau=})}{=}\tau.$

Finally we want to show that $R$ may be replaced by a nonempty subset $Y$ of $R$ . Clearly, the arguments in [Gub10, Proposition 5.7] generalize to the polystable situation, so we presume the claim for $K$ algebraically closed and show how to drop this assumption. Let $\mathbb{C}_{K}$ be the completion of an algebraic closure of $K$ . We denote by $\pi:\mathfrak{X}^{\prime\prime}_{\mathbb{C}_{K}}\rightarrow\mathfrak{X}^{\prime\prime}$ the base change of $\mathfrak{X}^{\prime\prime}$ to $\mathbb{C}_{K}^{\circ}$ . Let $R^{\prime}$ be the union of the strata of $\tilde{\mathfrak{X}}^{\prime\prime}_{\mathbb{C}_{K}}$ lying over $R$ . Then $\pi$ induces a surjection $p_{\mathfrak{X}^{\prime\prime}_{\mathbb{C}_{K}}}(\red^{-1}(R^{\prime}))\twoheadrightarrow p_{\mathfrak{X}^{\prime\prime}}(\red^{-1}(R))$ as the strata in $R^{\prime}$ correspond to open faces lying over $\tau$ . Let $Y^{\prime}$ be a lift of $Y$ in $R^{\prime}$ . By [Gub10, Proposition 5.7] we have $p_{\mathfrak{X}^{\prime\prime}_{\mathbb{C}_{K}}}(\red^{-1}(Y^{\prime}))=p_{\mathfrak{X}^{\prime\prime}_{\mathbb{C}_{K}}}(\red^{-1}(R^{\prime}))$ . Clearly $p_{\mathfrak{X}^{\prime\prime}}(\red^{-1}(Y))\subseteq p_{\mathfrak{X}^{\prime\prime}}(\red^{-1}(R))$ and hence it is enough to show that the restriction of $\pi$ to $p_{\mathfrak{X}^{\prime\prime}_{\mathbb{C}_{K}}}(\red^{-1}(Y^{\prime}))$ factors through $p_{\mathfrak{X}^{\prime\prime}}(\red^{-1}(Y))$ . We have the following commutative diagram:

Let $x\in p_{\mathfrak{X}^{\prime\prime}_{\mathbb{C}_{K}}}(\red^{-1}(Y^{\prime}))$ and $y\in\red^{-1}(Y^{\prime})$ with $p_{\mathfrak{X}^{\prime\prime}_{\mathbb{C}_{K}}}(y)=x$ then $\pi(x)=\pi(p_{\mathfrak{X}^{\prime\prime}_{\mathbb{C}_{K}}}(y))=p_{\mathfrak{X}^{\prime\prime}}(\pi(y))\in p_{\mathfrak{X}^{\prime\prime}}(\red^{-1}(Y))$ . This proves the claim. ∎

Corollary 2.9.

Let $R$ be a stratum of $\tilde{\mathfrak{X}}^{\prime\prime}$ corresponding to the open face $\tau$ of $\mathfrak{D}$ .

(a)

$\dim(\tau)=\codim(R,\tilde{\mathfrak{X}}^{\prime\prime})$ .
(b)

$S:=\tilde{\iota}(R)$ is a stratum of $\tilde{\mathfrak{X}}$ .
(c)

$R\overset{\tilde{\iota}}{\rightarrow}S$ is a fibre bundle with fibre $T$ where $T$ is the $\dim(R)-\dim(S)$ dimensional torus orbit from the proof of Proposition 2.8.
(d)

Every stratum of $\tilde{\mathfrak{X}}^{\prime\prime}$ is smooth.
(e)

The closure $\bar{R}$ is the union of all strata of $\tilde{\mathfrak{X}}^{\prime\prime}$ corresponding to open faces $\sigma$ of $\mathfrak{D}$ with $\tau\subseteq\bar{\sigma}$ .
(f)

For an irreducible component $Y$ of $\tilde{\mathfrak{X}}^{\prime\prime}$ , let $\zeta_{Y}$ be the unique point of $\mathfrak{X}^{\textup{an}}$ with reduction equal to the generic point of $Y$ . Then $Y\mapsto\zeta_{Y}$ is a bijection between the irreducible components of $\tilde{\mathfrak{X}}^{\prime\prime}$ and the vertices of $\mathfrak{D}$ .

Proof.

The statements can be proven the same way as in [Gub10, Corollary 5.9]. In order to bypass the algebraically closedness of the base field one can use [Gub13, Proposition 6.22] instead of [Gub07, Proposition 4.4] for (a), [Gub13, Proposition 6.22] instead of [Gub07, Remark 4.8] for (e) and [Gub13, Proposition 6.14] instead of [Gub07, Proposition 4.7] for (f). ∎

Definition 2.10.

Let $\Delta$ be a skeleton associated to a strongly nondegenerate strictly polystable formal scheme $\mathfrak{X}^{\prime}$ over $K^{\circ}$ . A continuous function $h:\Delta\rightarrow\mathbb{R}$ is called piecewise affine linear if there exists a $\Gamma$ -rational polytopal subdivision $\mathfrak{D}$ of $\Delta$ such that for any canonical polysimplex $\Delta_{S}$ of $\Delta$ , any formal open subset $\psi:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ of $\mathfrak{X}^{\prime}$ whose distinguished stratum is $S$ and any $\Delta^{\prime}\in\mathfrak{D}$ with $\Delta^{\prime}\subseteq\Delta_{S}$ , there exist $\boldsymbol{m}\in\mathbb{Z}^{\boldsymbol{n}+\boldsymbol{1}}$ and $\alpha\in K^{\times}$ such that $h\Big|_{\Delta^{\prime}}=(\boldsymbol{m}\cdot\boldsymbol{x}+v(\alpha))\circ\psi^{\textup{an}}\Big|_{\Delta^{\prime}}$ (see Definition 2.3 for the notation and setting).

Proposition 2.11.

Let $\mathfrak{X}^{\prime}$ be a strongly nondegenerate strictly polystable formal scheme over $K^{\circ}$ with associated skeleton $S(\mathfrak{X}^{\prime})$ and $h$ a piecewise affine linear function on $S(\mathfrak{X}^{\prime})$ . Let $\mathfrak{D}$ be a $\Gamma$ -rational polytopal subdivision of $S(\mathfrak{X}^{\prime})$ suitable for $h$ as in Definition 2.10 and $\iota:\mathfrak{X}^{\prime\prime}\rightarrow\mathfrak{X}^{\prime}$ be the canonical formal scheme over $\mathfrak{X}^{\prime}$ associated to $\mathfrak{D}$ (see Construction 2.6). Then $h$ induces a canonical Cartier divisor $D$ on $\mathfrak{X}^{\prime\prime}$ which is trivial on the generic fibre. If $\mathfrak{X}^{\prime\prime}$ is admissible, then $D$ has the property that $\|1\|_{\mathcal{O}(D)}=e^{-h\circ p_{\mathfrak{X}^{\prime}}}$ where $\|\cdot\|_{\mathcal{O}(D)}$ is the formal metric on $\mathcal{O}_{\mathfrak{X}^{\prime\textup{an}}}$ given by the formal model $\mathcal{O}(D)$ of $\mathcal{O}_{\mathfrak{X}^{\prime\textup{an}}}$ (see Definition 3.1).

Proof.

As in Construction 2.6, we cover $\mathfrak{X}^{\prime}$ by étale maps $\psi:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)=\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})\times\mathfrak{X}(m)$ and for each $\mathfrak{U}$ and $\Delta\in\mathfrak{D}$ with $\Delta\subseteq\mathfrak{U}^{\textup{an}}$ we obtain the affine formal scheme $\mathfrak{U}_{\Delta}$ . We write $\psi^{\prime}:\mathfrak{U}^{\prime\prime}_{\Delta}\rightarrow\mathfrak{U}_{\Delta}$ for the base change with respect to $\psi$ and obtain a cover of $\mathfrak{X}^{\prime\prime}$ . On $\Delta\in\mathfrak{D}$ , $h$ is given by $\boldsymbol{m}\boldsymbol{x}+v(\alpha)$ with $\boldsymbol{m}\in\mathbb{Z}^{\boldsymbol{n}+\boldsymbol{1}}$ , $\alpha\in K^{\times}$ . We define $D$ locally on $\mathfrak{U}^{\prime\prime}_{\Delta^{\prime}}$ by $\psi^{\prime\ast}(\alpha\cdot\boldsymbol{x}^{\boldsymbol{m}})$ . Then $D$ is indeed a Cartier Divisor on $\mathfrak{X}^{\prime\prime}$ as for $\mathfrak{U}_{1},\mathfrak{U}_{2},\Delta_{1},\Delta_{2}$ as above and $\mathfrak{U}:=\mathfrak{U}_{1}\cap\mathfrak{U}_{2}$ we have $\alpha_{1}\cdot\boldsymbol{x}^{\boldsymbol{m}_{1}}/\alpha_{2}\cdot\boldsymbol{x}^{\boldsymbol{m}_{2}}\in\mathcal{O}(\mathfrak{U}_{\Delta_{1}\cap\Delta_{2}})^{\times}$ since $\boldsymbol{m}_{1}\boldsymbol{x}+v(\alpha_{1})=\boldsymbol{m}_{2}\boldsymbol{x}+v(\alpha_{2})$ on $\Delta_{1}\cap\Delta_{2}$ . Hence

\displaystyle\psi_{1}^{\prime\ast}(\alpha_{1}\cdot\boldsymbol{x}^{\boldsymbol{m}_{1}})/\psi_{2}^{\prime\ast}(\alpha_{2}\cdot\boldsymbol{x}^{\boldsymbol{m}_{2}})\Big|_{\mathfrak{U}^{\prime\prime}_{\Delta_{1}\cap\Delta_{2}}}=\psi^{\prime\ast}(\alpha_{1}\cdot\boldsymbol{x}^{\boldsymbol{m}_{1}}/\alpha_{2}\cdot\boldsymbol{x}^{\boldsymbol{m}_{2}})\in\mathcal{O}(\mathfrak{U}^{\prime\prime}_{\Delta_{1}\cap\Delta_{2}})^{\times}

and therefore $\psi_{1}^{\prime\ast}(\alpha_{1}\boldsymbol{x}^{\boldsymbol{m}_{1}})/\psi_{2}^{\prime\ast}(\alpha_{2}\boldsymbol{x}^{\boldsymbol{m}_{2}})\in\mathcal{O}(\mathfrak{U}^{\prime\prime}_{1,\Delta_{1}}\cap\mathfrak{U}^{\prime\prime}_{2,\Delta_{2}})^{\times}$ . Furthermore $D$ is trivial on the generic fibre, as $\alpha\cdot\boldsymbol{x}^{\boldsymbol{m}}\in\mathcal{O}(\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\textup{an}})^{\times}$ . ∎

Remark 2.12.

Note that we can ensure that $\mathfrak{X}^{\prime\prime}$ is admissible and hence a formal model by performing base change to the completion of an algebraic closure of $K$ (see Construction 2.6) which will be enough for our purposes.

Lemma 2.13.

In the situation of Proposition 2.11 let $\tau$ be an open face of the skeleton $\Delta$ of dimension equal to the dimension of $\mathfrak{X}^{\prime an}$ and assume that $h$ is affine linear on $\bar{\tau}$ . Let $D$ be the induced Cartier divisor on $\mathfrak{X}^{\prime\prime}$ and $Y$ a proper curve in $\tilde{\mathfrak{X}}^{\prime\prime}$ with $Y\subseteq\red_{\mathfrak{X}^{\prime\prime}}(p_{\mathfrak{X}^{\prime\prime}}^{-1}(\tau))$ e.g. if $Y$ lies inside an irreducible component of $\tilde{\mathfrak{X}}^{\prime\prime}$ corresponding to a vertex $u\in\tau$ of $\mathfrak{D}$ . Then $\Deg(D.Y)=0$ .

Proof.

Note that we do not assume $\bar{\tau}\in\mathfrak{D}$ . But by passing to the formal open subscheme of $\mathfrak{X}^{\prime}$ consisting of the formal open subsets $\mathfrak{U}$ with $S(\mathfrak{U})=\bar{\tau}$ , we may assume $\Delta=\bar{\tau}$ and then the polytopal subdivision $\mathfrak{D}^{\prime}$ consisting the polytope $\bar{\tau}$ and its faces is suitable for $h$ . The corresponding formal scheme is $\mathfrak{X}^{\prime}$ . Let $D^{\prime}$ be the Cartier divisor on $\mathfrak{X}^{\prime}$ induced by $h$ as in Proposition 2.11. Notice that by construction we have $D=\iota^{\ast}D^{\prime}$ . Now $\iota$ is proper by [Tem00, Corollary 4.4] (the result requires $\mathfrak{X}^{\prime\prime}$ to be admissible but by [Gro65, Proposition 2.7.1] it is enough to check properness after base change to the completion of an algebraic closure of $K$ , after which $\mathfrak{X}^{\prime\prime}$ is always admissible, see Construction 2.6). Hence the projection formula yields $\Deg(D.Y)=\Deg(D^{\prime}.\iota_{\ast}Y)$ . Now

\iota(Y)\subseteq\iota(\red_{\mathfrak{X}^{\prime\prime}}(p_{\mathfrak{X}^{\prime\prime}}^{-1}(\tau)))=\red_{\mathfrak{X}^{\prime}}(p_{\mathfrak{X}^{\prime}}^{-1}(\tau)),

where the latter is the stratum in $\tilde{\mathfrak{X}}^{\prime}$ corresponding to $\tau$ and hence a point. Therefore $D^{\prime}.\iota_{\ast}Y=0$ . ∎