5. Descent for model functions [039C]

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5. Descent for model functions

Let $K$ denote a complete discretely valued field with valuation ring $K^{\circ}$ . Let $R$ be a discrete valuation subring of $K^{\circ}$ whose completion is $K^{\circ}$ . Then $K$ is the completion of the field of fractions $F$ of $R$ . In this section we show that all model functions on analytifications of varieties over $K$ are already defined over $R$ .

An $R$ -model of a projective variety over $F$ is defined completely analogously to the complete case treated in 2.1.

Definition 5.1.

Let $X$ be a projective variety over $K$ . We say that a model function $\varphi\colon X^{\mathrm{an}}\to\mathbb{R}$ is defined over $R$ if there exists a projective variety $Y$ over $F$ , with an isomorphism $Y\otimes_{F}K\simeq X$ , an $R$ -model ${\mathscr{Y}}$ of $Y$ , a vertical divisor $D_{0}$ on ${\mathscr{Y}}$ such that $\varphi=\frac{1}{m}\varphi_{D}$ where $m\in\mathbb{N}_{>0}$ and $D$ is the vertical divisor on ${\mathscr{Y}}\otimes_{R}K^{\circ}$ obtained by pullback from $D_{0}$ . Likewise we define the notion of a vertical ideal sheaf defined over $R$ .

Here is the announced descent result.

Proposition 5.2.

Let $Y$ be a projective variety over $F$ and let $X\coloneqq Y\otimes_{F}K$ .

(a)

Any $K^{\circ}$ -model of $X$ is dominated by the base change of a projective $R$ -model of $Y$ to $K^{\circ}$ .
(b)

If a projective $K^{\circ}$ -model ${{\mathscr{X}}}$ of $X$ dominates ${\mathscr{Y}}\otimes_{R}K^{\circ}$ for a projective $R$ -model ${\mathscr{Y}}$ of $Y$ , then ${{\mathscr{X}}}\simeq{\mathscr{Y}}^{\prime}\otimes_{R}K^{\circ}$ for a projective $R$ -model ${\mathscr{Y}}^{\prime}$ of $Y$ dominating ${\mathscr{Y}}$ .
(c)

Every model function on ${X^{{\mathrm{an}}}}$ is defined over $R$ .

Proof.

To prove (a), we pick any projective $R$ -model ${\mathscr{Y}}$ of $Y$ . By [Lü93, Lemma 2.2], there is a blowing up $\pi\colon{{\mathscr{X}}}^{\prime}\to{\mathscr{Y}}\otimes_{R}K^{\circ}$ such that ${{\mathscr{X}}}^{\prime}$ dominates ${{\mathscr{X}}}$ . Since $\pi$ is a projective morphism, ${{\mathscr{X}}}^{\prime}$ is a projective $K^{\circ}$ -model dominating ${{\mathscr{X}}}$ . Hence (a) follows from (b).

To prove (b), we note that the morphism ${{\mathscr{X}}}\to{\mathscr{Y}}\otimes_{R}K^{\circ}$ is a blowing up morphism along a vertical closed subscheme $Z$ of ${\mathscr{Y}}\otimes_{R}K^{\circ}$ (see [Liu06, Thm. 8.1.24]). Since the ideal sheaf of $Z$ contains a power of the uniformizer of $R$ , we may define it over $R$ and hence the same is true for the blowing up morphism and for ${{\mathscr{X}}}$ proving (b).

To prove (c), we may assume that the model function is associated to a vertical Cartier divisor $D$ . Replacing $D$ by $D+{\operatorname{div}}(\lambda)$ for a suitable non-zero $\lambda\in R$ and using (a) and (b), we may assume that $D$ is an effective Cartier divisor on a projective $R$ -model ${\mathscr{Y}}$ of $Y$ . As in (b), we see that the ideal sheaf of $D$ is defined by the ideal sheaf of a Cartier divisor $D_{0}$ defined over $R$ proving (c). ∎