3.3. Spectrum of normed section algebra [00N5]

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3.3. Spectrum of normed section algebra

We embed the Berkovich spectrum of normed section algebra into the analytification of the spectrum of the section algebra.

Lemma 3.4.

The homomorphism of inclusion of $k$ -algebras $V(L)\rightarrow\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi)$ induces a continuous map between topological spaces $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi))\rightarrow(\spec V(L))^{\mathrm{an}}$ which is closed. Moreover, the map is injective.

Proof.

This is clear by Proposition 2.88. ∎

Proposition 3.5.

There exist algebra norms $\vvvert\mathord{\cdot}\vvvert_{\phi}^{\mathrm{aff}}$ on $V_{{\scriptscriptstyle\bullet}}(L)$ and $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}^{\mathrm{aff}}$ on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ such that the separated completions of $(V_{{\scriptscriptstyle\bullet}}(L),\vvvert\mathord{\cdot}\vvvert_{\phi}^{\mathrm{aff}})$ and $(V_{{\scriptscriptstyle\bullet}}(L_{X|Y}),\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}^{\mathrm{aff}})$ are affinoid algebras. Denote them by $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi^{\mathrm{aff}})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}^{\mathrm{aff}})$ . Moreover, there exists a commutative diagram of homomorphisms of $k$ -algebras with $\sigma$ and $\sigma_{Y}$ being homomorphisms of Banach $k$ -algebras

All homomorphisms (except $i_{Y}$ ) have dense images, and $i_{Y}$ is surjective.

Proof.

As $V_{{\scriptscriptstyle\bullet}}(L)$ is a finitely generated sub- $k$ -algebra of $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi)$ which is dense for the topology induced by Banach algebra norm, by Proposition 2.44, there exists an affinoid algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}^{\mathrm{aff}}$ on $V_{{\scriptscriptstyle\bullet}}(L)$ and a homomorphism of Banach algebras

\sigma:\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi^{\mathrm{aff}})\rightarrow\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi).

One then takes $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}^{\mathrm{aff}}$ on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ to be the quotient norm of $\vvvert\mathord{\cdot}\vvvert_{\phi}^{\mathrm{aff}}$ . By Example 2.42, this quotient norm is also an affinoid algebra norm. By this quotient construction, there exists a homomorphism of Banach $k$ -algebras

\sigma|_{Y}:\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}^{\mathrm{aff}})\rightarrow\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})

which fits into a commutative diagram with $i_{Y}(\phi)$ and $i_{Y}(\phi^{\mathrm{aff}})$ . ∎

Corollary 3.6.

The commutative diagram of homomorphisms of $k$ -algebras induces a commutative diagram of continuous maps of topological spaces

All maps are closed. If the algebra seminorm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is a norm, then all maps are injective.

Proof.

This follows from Proposition 2.26, Proposition 2.88 and Proposition 2.27. ∎