ScalingStacks

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.