Proposition 3.16 (00KV) · ScalingStacks

00KV

Proposition 3.16. Let $z\in(\spec V_{{\scriptscriptstyle\bullet}}(L))^{\mathrm{an}}$ be a point. Let $(x,e^{\vee}(x))\in Tot(L^{\vee})$ be the point $(p(\boldsymbol{0})^{-1})^{\mathrm{an}}(z)$ . Then $z\in\mathfrak{M}(\widehat{V}(L,\vvvert\mathord{\cdot}\vvvert))$ if and only if one of the following criteria holds

(1)

there exist $C(z)>0$ such that

$\forall\underline{s}\in V_{{\scriptscriptstyle\bullet}}(L),\quad\lvert\underline{s}\rvert_{z}\leq C(z)\cdot\vvvert\underline{s}\vvvert.$

(2)

there exist $C(z)>0$ such that

\forall\underline{s}(x)\in V_{{\scriptscriptstyle\bullet}}(L)(x),\quad\lvert\underline{s}(x)\rvert_{z}\leq C(z)\cdot\vvvert\underline{s}(x)\vvvert_{X|x}.

(3)

there exist $C^{\prime}(z)=1$ such that

$\forall e_{1}(x)\in V_{1}(L)(x),\quad\lvert e_{1}(x)\rvert_{z}\leq\vvvert e_{1}(x)\vvvert_{(X|x);\mathrm{sp}},$

where $\vvvert\mathord{\cdot}\vvvert_{(X|x);\mathrm{sp}}$ is the spectral algebra seminorm of $\vvvert\mathord{\cdot}\vvvert_{X|x}$ .

00KW

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. ∎