ScalingStacks

Proof. By assumption, $L^{\otimes M}$ is very ample. So $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L,\phi)$, $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L_{X|Y},\phi_{X|Y})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L|_{Y},\phi|_{Y})$ are affinoid algebras. Since $V_{{\scriptscriptstyle\bullet}}(L)$ is integral and is finite over $V_{{\scriptscriptstyle\bullet}}^{(M)}(L)$, by Proposition 2.45, the Banach algebras $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi)$, $\widehat{V}(L_{X|Y},\phi_{X|Y})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}(L|_{Y},\phi|_{Y})$ are Banach finite over $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L,\phi)$, $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L_{X|Y},\phi_{X|Y})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L|_{Y},\phi|_{Y})$ respectively. Hence they are affinoid algebras. ∎