ScalingStacks

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. (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. (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. (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 $\vvvert\mathord{\cdot}\vvvert_{X|x}$.