ScalingStacks

Proof. Let $\{a_{i}\}_{i\in\{1,\dots,m\}}$ be a set of generators of $A$. For each $i\in\{1,\dots,m\}$, let $r_{i}$ denote $\vvvert a_{i}\vvvert_{\mathcal{B}}$ and let $\boldsymbol{r}$ denote the multi-radius consisting of $\{r_{i}\}_{i\in\{1,\dots,m\}}$. Consider the Tate algebra $\mathcal{T}_{\boldsymbol{r}}$ and the homomorphism of $k$-algebras

$$k[T_{1},\dots,T_{m}]\rightarrow\mathcal{B},\quad T_{i}\mapsto a_{i}$$

By the ultra-metricity of $\vvvert\mathord{\cdot}\vvvert_{\mathcal{B}}$ and the definition of $\boldsymbol{r}$, one has

$$\forall n\in\mathbb{N},\forall J\in\mathbb{N}^{m},\forall f_{J}\in k,\vvvert\sum_{J\in\mathbb{N}^{m}}f_{J}\cdot\boldsymbol{a}^{J}\vvvert_{\mathcal{B}}\leq\vvvert\sum_{J\in\mathbb{N}^{m}}f_{J}\cdot\boldsymbol{a}^{J}\vvvert_{\mathcal{T}_{\boldsymbol{r}}}$$

so by a density argument one can extend it to a homomorphism of Banach $k$-algebras

$$\mathcal{T}_{\boldsymbol{r}}\rightarrow\mathcal{B},\quad T_{i}\mapsto f_{i}$$

Let $\mathscr{I}$ be the kernel ideal of this homomorphism. To conclude it suffices to take $\mathcal{A}$ as $\mathcal{T}_{\boldsymbol{r}}/\mathscr{I}$. ∎