ScalingStacks

Proposition 2.11 . [059N]

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Proposition 2.11.

Let 𝔛′\mathfrak{X}^{\prime} be a strongly nondegenerate strictly polystable formal scheme over K∘K^{\circ} with associated skeleton S⁑(𝔛′)S(\mathfrak{X}^{\prime}) and hh a piecewise affine linear function on S⁑(𝔛′)S(\mathfrak{X}^{\prime}). Let 𝔇\mathfrak{D} be a Ξ“\Gamma-rational polytopal subdivision of S⁑(𝔛′)S(\mathfrak{X}^{\prime}) suitable for hh as in Definition 2.10 and ΞΉ:𝔛′′→𝔛′\iota:\mathfrak{X}^{\prime\prime}\rightarrow\mathfrak{X}^{\prime} be the canonical formal scheme over 𝔛′\mathfrak{X}^{\prime} associated to 𝔇\mathfrak{D} (see Construction 2.6). Then hh induces a canonical Cartier divisor DD on 𝔛′′\mathfrak{X}^{\prime\prime} which is trivial on the generic fibre. If 𝔛′′\mathfrak{X}^{\prime\prime} is admissible, then DD has the property that βˆ₯1βˆ₯π’ͺ⁑(D)=eβˆ’h∘p𝔛′\|1\|_{\mathcal{O}(D)}=e^{-h\circ p_{\mathfrak{X}^{\prime}}} where βˆ₯β‹…βˆ₯π’ͺ⁑(D)\|\cdot\|_{\mathcal{O}(D)} is the formal metric on π’ͺ𝔛′an\mathcal{O}_{\mathfrak{X}^{\prime\textup{an}}} given by the formal model π’ͺ⁑(D)\mathcal{O}(D) of π’ͺ𝔛′an\mathcal{O}_{\mathfrak{X}^{\prime\textup{an}}} (see Definition 3.1).

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