4.1. Localization of spectrum by affinoid domain covering [00NA]

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4.1. Localization of spectrum by affinoid domain covering

For any $\epsilon>0$ , one can identify $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon)))$ with $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}))$ by Corollary 3.27. Via this identification, one has

\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\subseteq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))).

By Corollary 3.6, they can be both identified with closed subsets in $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$ , which itself can be identified with a closed subset in $\spec(V_{{\scriptscriptstyle\bullet}}(L_{X|Y}))^{\mathrm{an}}$ . We shall work in this fixed affinoid domain. We use notations $\text{Int}^{\mathrm{top}}_{\mathfrak{M}}$ and $\text{Int}^{\mathrm{top}}_{V_{{\scriptscriptstyle\bullet}}}$ to distinguish the topological interior in $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$ and in $\spec(V_{{\scriptscriptstyle\bullet}}(L_{X|Y}))^{\mathrm{an}}$ .

Lemma 4.1.

Assume that $\phi|_{Y}$ is asymptotic Fubini-Study. For any $\epsilon>0$ , $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$ is contained in $\text{Int}^{\mathrm{top}}_{\mathfrak{M}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))))$ .

Proof.

By Proposition 3.26, $\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}})$ is equal to $\phi|_{Y}$ , so it is continuous. Hence for any $\epsilon>0$ , by Proposition 3.23, one has

\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))=\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}))\subseteq\text{Int}^{\mathrm{top}}_{V_{{\scriptscriptstyle\bullet}}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon)))).

By Proposition 2.88, the topology on $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$ coincides the induced topology from $\spec(V_{{\scriptscriptstyle\bullet}}(L|_{Y}))^{\mathrm{an}}$ , hence the set $\text{Int}^{\mathrm{top}}_{V_{{\scriptscriptstyle\bullet}}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))))$ which is open in $\spec(V_{{\scriptscriptstyle\bullet}}(L|_{Y}))^{\mathrm{an}}$ is also open in $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$ . Therefore this set is contained in $\text{Int}^{\mathrm{top}}_{\mathfrak{M}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))))$ . ∎

Proposition 4.2.

Assume that $\phi|_{Y}$ is asymptotic Fubini-Study. For any $\epsilon>0$ , there exist a special domain $W_{\epsilon}$ such that

\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\subseteq W_{\epsilon}\subseteq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))).

Proof.

By Corollary 2.64, every point in $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$ has a neighbourhood system consisting of affinoid domains. Hence for any $z\in\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$ , there exists an affinoid domain neighbourhood $V(z)$ . By Lemma 4.1, $z$ has an open neighbourhood $\text{Int}^{\mathrm{top}}_{\mathfrak{M}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))))$ , so we can assume that each $V(z)$ is contained in this open set.

One forms a covering by open sets

\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\subseteq\bigcup_{z\in\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))}\text{Int}^{\mathrm{top}}_{\mathfrak{M}}V(z).

Since the left hand side is a compact set by Proposition 2.21, there exist finitely many points $\{z_{1},\dots,z_{m}\}$ such that $\{\text{Int}^{\mathrm{top}}_{\mathfrak{M}}V(z_{i})\}_{i\in\{1,\dots,m\}}$ form a covering of $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$ . Let $W_{\epsilon}$ be the union of affinoid domains $\{V(z_{i})\}_{i\in\{1,\dots,m\}}$ , then it is a special domain, and satisfies the desired inclusion conditions. ∎

Remark 4.3.

One denotes by $\mathcal{W}_{\epsilon}$ the Banach $k$ -algebra of $\Gamma(W_{\epsilon},\mathscr{O}_{W_{\epsilon}})$ equipped with supremum norm $\vvvert\mathord{\cdot}\vvvert_{W_{\epsilon}}$ (see Definition 2.70).