Subsection 3.3. (00N5) · ScalingStacks

00K8

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.

00K9

Proof. This is clear by Proposition 2.88. ∎

00KA

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.

00KB

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 $i_{Y}(\phi^{\mathrm{aff}})$ . ∎

00KC

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.

00KD

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

3.3. Spectrum of normed section algebra