Appendix A Reduction of germs [05BR]

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Appendix A Reduction of germs

In this appendix we will explain the reduction of germs due to Michael Temkin (see [Tem00] and [Tem04]). At the end we will use this theory to prove a generalization of [CD, Lemme 6.5.1] proposed by Antoine Ducros which drops a separatedness assumption.

Definition A.1.

i)

The category of punctual strictly $K$ -analytic spaces is the following: The objects are pairs $(X,x)$ where $X$ is a strictly $K$ -analytic space and $x\in X$ is a point. A morphism $\varphi:(X,x)\rightarrow(Y,y)$ is a morphism $\varphi:X\rightarrow Y$ of strictly $K$ -analytic spaces such that $\varphi(x)=y$ .
ii)

The category $(K\textrm{-Germs})$ of germs of a strictly $K$ -analytic space at a point is defined to be the localization of the category of punctual strictly $K$ -analytic spaces by the system of morphisms $\varphi:(X,x)\rightarrow(Y,y)$ which identify $X$ with an open neighbourhood of $y$ in $Y$ . The germ induced by the punctual strictly $K$ -analytic space $(X,x)$ is denoted by $X_{x}$ .
iii)

A germ $X_{x}$ is said to be good if $x$ has a strictly $K$ -affinoid neighbourhood in $X$ . A morphism of germs $\varphi:X_{x}\rightarrow Y_{y}$ is said to be separated resp. closed if it is induced by a separated resp. boundaryless morphism $X^{\prime}\rightarrow Y$ for an open neighbourhood $X^{\prime}$ of $x$ in $X$ (recall that a morphism $\varphi:X\rightarrow Y$ of $K$ -analytic spaces is called boundaryless if $X=\Int(X/Y)$ , where the relative interior $\Int(X/Y)$ is defined to be the set of all $x\in X$ such that for any affinoid domain $V\subseteq Y$ with $\varphi(x)\in V$ there is an affinoid neighbourhood $U\subseteq\varphi^{-1}(V)$ of $x$ in $\varphi^{-1}(V)$ such that $x\in\Int(U/V)$ ).

Definition A.2.

Let $k$ be a field and let $L$ be a field extension of $k$ .

i)

The Zariski-Riemann space $\boldsymbol{P}_{L/k}$ is the set of valuation rings in $L$ which contain $k$ and whose quotient field is $L$ endowed with the coarsest topology such that all sets of the form $\boldsymbol{P}_{L/k}\{f\}:=\left\{R\in\boldsymbol{P}_{L/k}\;\Big|\;f\in R\right\}$ with $f\in L$ are open.
ii)

The category $(\textrm{bir}_{k})$ is the following: The objects are triples $(X,L,\phi)$ where $X$ is a connected quasi-compact and quasi-separated topological space, $L$ is a field extension of $k$ and $\phi:X\rightarrow\boldsymbol{P}_{L/k}$ is a local homeomorphism. A morphism $(X,L,\phi)\rightarrow(Y,M,\psi)$ is a pair $(h,i)$ where $h:X\rightarrow Y$ is a continuous map and $i:M\rightarrow L$ is a morphism of field extensions of $k$ such that $\psi\circ h=i^{\#}\circ\phi$ where $i^{\#}:\boldsymbol{P}_{L/k}\rightarrow\boldsymbol{P}_{M/k}$ is the morphism induced by $i$ .
iii)

A morphism $(h,i):(X,L,\phi)\rightarrow(Y,M,\psi)$ is called proper if the map $X\rightarrow Y\times_{\boldsymbol{P}_{M/k}}\boldsymbol{P}_{L/k}$ is bijective.

In [Tem00, §2] Temkin introduced a reduction functor $\red$ from $(K\textrm{-Germs})$ to $(\textrm{bir}_{\tilde{K}})$ sending a germ $X_{x}$ to its reduction $\tilde{X_{x}}$ . It can be described as follows (see [Tem04, §4]): If $X_{x}$ is a good germ, we can assume $X=\mathscr{M}(A)$ for a strictly $K$ -affinoid algebra $A$ . Then the character $\chi_{x}:A\rightarrow\mathscr{H}(x)$ induces a morphism $\tilde{\chi_{x}}:\tilde{A}\rightarrow\widetilde{\mathscr{H}(x)}$ . Then $\tilde{X_{x}}=(\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}\{\tilde{\chi_{x}}(\tilde{A})\},\widetilde{\mathscr{H}(x)},\iota)$ where $\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}\{\tilde{\chi_{x}}(\tilde{A})\}$ is the set of all $R\in\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ for which $\tilde{\chi}(\tilde{A})\subseteq R$ and $\iota$ is the canonical embedding. If $X_{x}$ is separated one covers $X_{x}$ by finitely many good germs $V_{x}^{i}$ . Then the germs $V_{x}^{i}\cap V_{x}^{j}$ are good and one obtains an open embedding $\widetilde{V_{x}^{i}\cap V_{x}^{j}}\rightarrow\tilde{V_{x}^{i}}$ . In fact this gives a glueing data and $\tilde{X}_{x}$ is the space obtained by glueing the $\tilde{V_{x}^{i}}$ along these open embeddings. Lastly if $X_{x}$ is arbitrary, one covers $X_{x}$ by finitely many separated germs $V_{x}^{i}$ and again gets open embeddings $\widetilde{V_{x}^{i}\cap V_{x}^{j}}\rightarrow\tilde{V_{x}^{i}}$ along which the $\tilde{V_{x}^{i}}$ are glued to $\tilde{X_{x}}$ .

Proposition A.3.

Let $\mathfrak{X}$ be an admissible formal scheme and $x\in X:=\mathfrak{X}^{\textup{an}}$ . Let $V$ be the closure of $\{\red(x)\}$ in the special fibre $\tilde{\mathfrak{X}}$ . Then $V$ is proper if and only if the morphism $\tilde{X_{x}}\rightarrow\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ is bijective.

Proof.

Let $(Y_{i})_{i\in I}$ be an open affine cover of $V$ and set $V^{i}:=\red^{-1}(Y_{i})$ . Then $V^{i}$ is strictly $K$ -affinoid by [Bos77, Theorem 3.1] and hence $V^{i}_{x}$ is a good germ. Note that $(V^{i}_{x})_{i\in I}$ is a cover of $X_{x}$ . Hence $\tilde{X_{x}}$ is obtained by glueing the $\tilde{V^{i}_{x}}$ along the canonical maps $\widetilde{V^{i}_{x}\cap V^{j}_{x}}\rightarrow\tilde{V^{i}_{x}}$ . Let $V^{i}=\mathscr{M}(A_{i})$ for a strictly $K$ -affinoid algebra $A_{i}$ . Then $Y_{i}=\Spec{\tilde{A_{i}}}$ and the character $\chi_{x}:A_{i}\rightarrow\mathscr{H}(x)$ induces a morphism $\tilde{\chi_{x}}:\tilde{A_{i}}\rightarrow\widetilde{\mathscr{H}(x)}$ . Let $\mathfrak{p}\subseteq\tilde{A_{i}}$ be the prime ideal corresponding to $\red(x)$ i.e. $\mathfrak{p}$ is the kernel of $\tilde{\chi_{x}}$ . The induced morphism $\tilde{A_{i}}/\mathfrak{p}\rightarrow\widetilde{\mathscr{H}(x)}$ is injective and hence it extends to a morphism $\tilde{K}(V)\rightarrow\widetilde{\mathscr{H}(x)}$ where $\tilde{K}(V)=\Quot(\tilde{A_{i}}/\mathfrak{p})$ denotes the function field of $V$ . This induces a morphism $\pi:\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}\rightarrow\boldsymbol{P}_{\tilde{K}(V)/\tilde{K}}$ .
First step: We have that $\tilde{V^{i}_{x}}=\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}\{\tilde{\chi_{x}}(\tilde{A_{i}})\}$ is the preimage under $\pi$ of the set of valuation rings in $\tilde{K}(V)$ which admit a center on $Y_{i}\cap V$ .
Indeed if $R\in\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ is a valuation ring with $\tilde{\chi_{x}}(\tilde{A_{i}})\subseteq R$ then $\tilde{A_{i}}/\mathfrak{p}\subseteq R\cap\tilde{K}(V)$ . Let $\mathfrak{m}_{R}$ be the maximal ideal of $R$ then $\mathfrak{p}^{\prime}:=\mathfrak{m}_{R}\cap\tilde{A_{i}}/\mathfrak{p}$ defines a point in $\Spec(\tilde{A_{i}}/\mathfrak{p})$ whose local ring is $(\tilde{A_{i}}/\mathfrak{p})_{\mathfrak{p}^{\prime}}$ and we have $(\tilde{A_{i}}/\mathfrak{p})_{\mathfrak{p}^{\prime}}\subseteq R$ . Then $R\cap\tilde{K}(V)$ admits the center $\mathfrak{p}^{\prime}$ on $Y_{i}\cap V$ as claimed. Conversely if $R\cap\tilde{K}(V)$ admits a center on $Y_{i}\cap V$ then there exists $\mathfrak{p}^{\prime}\in\Spec(\tilde{A_{i}}/\mathfrak{p})$ such that $(\tilde{A_{i}}/\mathfrak{p})_{\mathfrak{p}^{\prime}}\subseteq R\cap\tilde{K}(V)$ and hence obviously $\tilde{\chi_{x}}(\tilde{A_{i}})\subseteq R$ .
Second step: The map $\tilde{X_{x}}\rightarrow\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ is surjective if and only if any valuation on $\tilde{K}(V)/\tilde{K}$ admits at least one center on $V$ .
Let $\tilde{X_{x}}\rightarrow\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ be surjective and $v$ a valuation on $\tilde{K}(V)/\tilde{K}$ . Then $v$ extends to a valuation $\tilde{v}$ on $\widetilde{\mathscr{H}(x)}$ . Let $R$ be the valuation ring of $\tilde{v}$ . Then $R\in\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ and hence $R$ has a preimage $R^{\prime}\in\tilde{X_{x}}$ . Then there exists $i\in I$ such that $R^{\prime}\in\tilde{V^{i}_{x}}$ hence the image of $R^{\prime}$ in $\boldsymbol{P}_{\tilde{K}(V)/\tilde{K}}$ admits a center on $Y_{i}\cap V$ by the first step. But this image is $R\cap\tilde{K}(V)$ by construction which is the valuation ring of $v$ . Hence $v$ admits a center on $V$ . Conversely suppose that any valuation on $\tilde{K}(V)/\tilde{K}$ admits a center on $V$ and let $R\in\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ then the image of $R$ in $\boldsymbol{P}_{\tilde{K}(V)/\tilde{K}}$ induces a valuation on $\tilde{K}(V)/\tilde{K}$ which admits a center $z$ on $V$ . Let $i\in I$ such that $z\in Y_{i}$ then $R\in\tilde{V^{i}_{x}}$ and the induced element in $\tilde{X_{x}}$ is a preimage of $R$ .
Third step: The map $\tilde{X_{x}}\rightarrow\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ is injective if and only if every valuation on $\tilde{K}(V)/\tilde{K}$ admits at most one center on $V$ .
To see this we describe $\widetilde{V^{i}_{x}\cap V^{j}_{x}}$ . In order to do so we cover $Y_{i}\cap Y_{j}$ by open affine subsets $Y_{i,j}^{k}$ . Their preimages under $\red$ yield a cover of $V^{i}_{x}\cap V^{j}_{x}$ by good germs. As above their reductions can be described as the preimage of the set of valuation rings in $\tilde{K}(V)$ which admit a center on $Y_{i,j}^{k}\cap V$ . The reduction of $V^{i}_{x}\cap V^{j}_{x}$ is then obtained by glueing these spaces. Now suppose that any valuation on $\tilde{K}(V)/\tilde{K}$ admits at most one center and let $R_{1},R_{2}\in\tilde{X_{x}}$ which map to the same valuation ring $R\in\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ . There exists $i,j$ such that $R_{1}\in\tilde{V^{i}_{x}},R_{2}\in\tilde{V^{j}_{x}}$ . As we have seen in the first step, $R_{1}\cap\tilde{K}(V)$ and $R_{2}\cap\tilde{K}(V)$ admit centers $y_{1}\in\Spec(\tilde{A_{i}})\cap V$ respectively $y_{2}\in\Spec(\tilde{A_{j}})\cap V$ . Then both are a center of $R\cap\tilde{K}(V)$ . Hence $y_{1}=y_{2}\in Y_{i}\cap Y_{j}$ by our assumption. Therefore by the first step $R_{1}=R_{2}=R$ in $\widetilde{V^{i}_{x}\cap V^{j}_{x}}$ . Hence in the glueing process, $R_{1}$ and $R_{2}$ are identified with each other. Conversely suppose that there is a valuation on $\tilde{K}(V)/\tilde{K}$ which admits two centers $y_{1},y_{2}\in V$ . Let $y_{1}\in Y_{i}$ and $y_{2}\in Y_{j}$ . Choose an extension of the valuation to $\widetilde{\mathscr{H}(x)}$ and let $R$ denote its valuation ring. Then $R$ induces an element $R_{1}\in\tilde{V^{i}_{x}}$ as well as an element $R_{2}\in\tilde{V^{j}_{x}}$ . Then $R_{1}$ and $R_{2}$ map to the same element $R$ in $\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ but they are not identified in the glueing process as $Y_{i}\cap V$ and $Y_{j}\cap V$ are separated and hence $R_{1}$ and $R_{2}$ admit at most one center in $\Spec(\tilde{A_{i}})\cap V$ respectively $\Spec(\tilde{A_{j}})\cap V$ which means in particular that they do not admit a center in $Y_{i}\cap Y_{j}\cap V$ . Hence $\tilde{X_{x}}\rightarrow\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ is not injective. This proves the third step.
Recall that $V$ is proper if and only if every valuation on $\tilde{K}(V)/\tilde{K}$ admits a unique center on $V$ ([Har77, Ch. II, Ex. 4.5]). Hence the claim follows from the second and third step. ∎

Corollary A.4.

In the situation of Proposition A.3, $x$ is an interior point of $X$ if and only if $V$ is proper.

Proof.

By Proposition A.3, $V$ is proper if and only if the map $\tilde{X_{x}}\rightarrow\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ is bijective which by [Tem04, Theorem 5.2] is equivalent to the map $X_{x}\rightarrow\mathscr{M}(K)$ being closed. But this is equivalent to $x$ being an interior point of $X$ . ∎