ScalingStacks

Proof. Let $\{c_{i}\}_{i\in\{1,\dots,m\}}\subset\mathcal{C}$ be a finite set of generators of $\mathcal{C}$ over $\mathcal{A}$, then consider an $\mathcal{A}$-Tate algebra $\mathcal{A}\{\boldsymbol{r}^{-1}\boldsymbol{T}\}$ where $r_{i}\geq\vvvert c_{i}\vvvert_{\mathcal{C}}$. There is a surjective $k$-algebra homomorphism defined by

$$\gamma:\mathcal{A}\{\boldsymbol{r}^{-1}\boldsymbol{T}\}\to\mathcal{C},\text{ }T_{i}\mapsto c_{i}$$

which is bounded as there exists $C>0$ such that

$$\vvvert\gamma(\sum_{J}a_{J}\boldsymbol{T}^{J})\vvvert_{\mathcal{C}}\leq\max_{J\in\mathbb{N}^{m}}\vvvert a_{J}\boldsymbol{c}^{J}\vvvert_{\mathcal{C}}\leq C\max_{J\in\mathbb{N}^{m}}\vvvert a_{J}\vvvert_{\mathcal{A}}\cdot\boldsymbol{r}^{J}$$

By Corollary 2.5, $\gamma$ is admissible, the norm $\vvvert\mathord{\cdot}\vvvert_{\mathcal{C}}$ is equivalent to the quotient norm of the $\mathcal{A}$-Tate norm. Hence $\mathcal{C}$ is an affinoid algebra. The strictness is obtained by choosing $r_{i}\in\lvert k^{\times}\rvert$ (see Lemma 2.47). ∎