ScalingStacks

Proof. The criterion 1 unfolds the definition of the fact that $z\in\mathfrak{M}$. The criterion 2 is equivalent to the criterion 1, as $\vvvert\mathord{\cdot}\vvvert_{X|x}$ is the quotient algebra norm of $\vvvert\mathord{\cdot}\vvvert_{\widehat{\kappa}(x)}$ for the evaluation map $\mathrm{ev}(x)$. The criterion 3 is equivalent to the criterion 2: if 2 holds, then

$$\forall n\in\mathbb{N},\quad\lvert e_{1}(x)\rvert_{z}\leq C(z)^{\frac{1}{n}}\cdot\vvvert e_{1}^{\otimes n}(x)\vvvert_{X|x}^{\frac{1}{n}},$$

so 3 holds after a limit process for $n\to\infty$. Conversely, if 3 holds, then since $V_{n}(L)(x)$ is spaned by $e_{1}^{\otimes n}(x)$ over $\widehat{\kappa}(x)$, one has

$$\forall n\in\mathbb{N},\quad\lvert s_{n}\rvert_{z}\leq\vvvert s_{n}(x)\vvvert_{(X|x);\mathrm{sp}}\leq\vvvert s_{n}(x)\vvvert_{X|x},$$

so 2 holds by the ultra-metricity of $\lvert\mathord{\cdot}\rvert_{z}$ and the orthogonality of $\vvvert\mathord{\cdot}\vvvert_{X|x}$ for $V_{n}(L)$’s. ∎