Proposition 3.14 (00KR) · ScalingStacks

00KR

Proposition 3.14. Assume that $L^{\otimes n}$ is globally generated. Let $\{s_{n,j}\}_{j\in\{0,\dots,d_{n}\}}$ be a basis of $V_{n}(L)$ . Let $\lVert\mathord{\cdot}\rVert_{n}$ be a ultrametric norm on $V_{n}(L)$ with respect to which $\{s_{n,j}\}_{j\in\{0,\dots,d_{n}\}}$ is orthogonal. Then for any $x\in X^{\mathrm{an}}$ and $e_{n}(x)\in L^{\otimes n}(x)\setminus 0$ ,

\lvert e_{n}(x)\rvert_{\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})}=\min_{j\in\{0,\dots,d_{n}\}}\Big\{\Big|\frac{e_{n}(x)}{s_{n,j}(x)}\Big|_{\widehat{\kappa}(x)}\cdot\lVert s_{n,j}\rVert_{n}\Big\},

with the convention that $0^{-1}=+\infty$ . (see also [CMor18, Lemma 3.3])

00KS

Proof. For $j\in\{0,\dots,d_{n}\}$ , let $\lambda_{j}\in\widehat{\kappa}(x)$ be such that $s_{n,j}(x)=\lambda_{j}\cdot e_{n}(x)$ . By construction,

\begin{split}\lvert e_{n}(x)\rvert_{\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})}=&\inf_{\mu_{j}\in\widehat{\kappa}(x),\ \sum\mu_{j}\cdot s_{n,j}(x)=e_{n}(x)}\Big\|\sum_{j\in\{0,\dots,d_{n}\}}\mu_{j}\cdot s_{n,j}\Big\|\\ =&\inf_{\mu_{j}\in\widehat{\kappa}(x),\ \sum\mu_{j}\cdot s_{n,j}(x)=e_{n}(x)}\max_{j}\Big\{\lvert\mu_{j}\rvert_{\widehat{\kappa}(x)}\cdot\lVert s_{n,j}\rVert\Big\}\\ =&\inf_{\kappa_{j}\in\widehat{\kappa}(x),\ \sum\kappa_{j}=1}\max_{j}\Big\{\lvert\kappa_{j}\cdot\lambda_{j}^{-1}\rvert_{\widehat{\kappa}(x)}\cdot\lVert s_{n,j}\rVert\Big\}\\ =&\inf_{\kappa_{j}\in\widehat{\kappa}(x),\ \sum\kappa_{j}=1}\max_{j}\Big\{\lvert\kappa_{j}\rvert_{\widehat{\kappa}(x)}\cdot\lvert\lambda_{j}\rvert_{\widehat{\kappa}(x)}^{-1}\cdot\lVert s_{n,j}\rVert\Big\}\\ =&\min_{j\in\{0,\dots,d_{n}\}}\Big\{\lvert\lambda_{j}\rvert_{\widehat{\kappa}(x)}^{-1}\cdot\lVert s_{n,j}\rVert_{n}\Big\}.\end{split}

The last equality is obtained with Lemma 3.13. ∎