Corollary 3.23 (00L8) · ScalingStacks

Corollary 3.23. If $\mathcal{P}(\phi)$ is continous, then for any $\epsilon>0$ , one has

\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi))\subseteq\text{Int}^{\mathrm{top}}_{V_{{\scriptscriptstyle\bullet}}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi(\epsilon)))),

where $\text{Int}^{\mathrm{top}}_{V_{{\scriptscriptstyle\bullet}}}$ denotes the topological interior as subspace of $\spec(V_{{\scriptscriptstyle\bullet}}(L))^{\mathrm{an}}$ .

00L9

Proof. By Proposition 3.20, the left hand side $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi))$ is identified with $p(\boldsymbol{0})^{\mathrm{an}}(\overline{\mathbb{D}}^{\vee}(L,\phi))$ , which is contained in the open subset $p(\boldsymbol{0})^{\mathrm{an}}(\mathbb{D}^{\vee}(L,\phi(\epsilon)))$ . This open subset is contained in $p(\boldsymbol{0})^{\mathrm{an}}(\overline{\mathbb{D}}^{\vee}(L,\phi(\epsilon)))$ which is identified with $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi(\epsilon)))$ , hence this open subset is contained in the topological interior of the later, the right hand side. ∎