ScalingStacks

Proposition 3.14 . [00KR]

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Proposition 3.14.

Assume that L⊗nL^{\otimes n} is globally generated. Let {sn,j}j∈{0,…,dn}\{s_{n,j}\}_{j\in\{0,\dots,d_{n}\}} be a basis of Vn​(L)V_{n}(L). Let ∥⋅∥n\lVert\mathord{\cdot}\rVert_{n} be a ultrametric norm on Vn​(L)V_{n}(L) with respect to which {sn,j}j∈{0,…,dn}\{s_{n,j}\}_{j\in\{0,\dots,d_{n}\}} is orthogonal. Then for any x∈Xanx\in X^{\mathrm{an}} and en​(x)∈L⊗n​(x)∖0e_{n}(x)\in L^{\otimes n}(x)\setminus 0,

|en​(x)|FS⁡(∥⋅∥n)=minj∈{0,…,dn}⁡{|en​(x)sn,j​(x)|κ^​(x)⋅∥sn,j∥n},\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=+∞0^{-1}=+\infty. (see also [CMor18, Lemma 3.3])

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