Subsection 5.1.1. (00NE) · ScalingStacks

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5.1.1. Case for $(\mathbb{P}^{d},\mathscr{O}(1))$

Let $\phi$ be a metric on $\mathscr{O}(1)$ , one studies the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ on $V_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1))$ . One would like to show that with various assumptions, it is a Gauss algebra norm, namely the standard affinoid algebra norm on the polynomial algebra. Then the normed section algebra $(V_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1)),\vvvert\mathord{\cdot}\vvvert_{\phi})$ will be a Tate affinoid algebra. (see Definition 2.38)

By Proposition 2.15, there exist an orthogonal basis $\{T_{i}\}_{i\in\{0,\dots,d\}}$ for the normed vector space $(V_{1}(\mathscr{O}(1)),\lVert\mathord{\cdot}\rVert_{\phi})$ . For any $i\in\{0,\dots,d\}$ , one denotes by $r_{i}$ the value $\lVert T_{i}\rVert_{\phi}$ , and by $\boldsymbol{r}\in(\mathbb{R}_{+})^{d+1}$ the multi-radius $(r_{0},\dots,r_{d})$ . One fixes such an orthogonal basis, and identify the graded $k$ -algebra $V_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1))$ with $k[T_{0},\dots,T_{d}]$ . For any multi-index $J=(j_{0},\dots,j_{d})\in\mathbb{N}^{d+1}$ , one denotes by $\boldsymbol{T}^{J}$ the monomial element $\prod_{i\in\{0,\dots,d\}}(T_{i})^{j_{i}}\in V_{\lvert J\rvert}(\mathscr{O}(1))$ .

Note that in general, the sub-spaces $V_{n}(\mathscr{O}(1))$ are orthogonal with respect to $\vvvert\mathord{\cdot}\vvvert_{\phi}$ for different $n\in\mathbb{N}$ , while a Gauss algebra norm exhibits a much finer orthogonality: the sub-spaces generated by each mononial $\boldsymbol{T}^{J}$ should be orthogonal for different $J\in\mathbb{N}^{d+1}$ .

First, on monomial elements, the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ resembles a Gauss norm.

00LV

Proposition 5.1. For any $J\in\mathbb{N}^{d+1}$ , one has

\lVert\boldsymbol{T}^{J}\rVert_{|J|\phi}=\prod_{i\in\{0,\dots,d\}}\lVert T_{i}\rVert_{\phi}^{j_{i}}.

00LW

Proof. Take a complete non-Archimedean valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ of $(k,\lvert\mathord{\cdot}\rvert_{k})$ such that

\forall i\in\{0,\dots,d\},\quad\{r_{i}\}_{i\in\{0,\dots,d\}}\subseteq\lvert k^{\times}\rvert_{K},

hence for any $i\in\{0,\dots,d\}$ , there exist elements $\kappa_{i}\in K$ such that $\lvert\kappa_{i}\rvert_{K}=r_{i}$ . One denotes by $x(\boldsymbol{r})\in(\mathbb{P}^{d}_{k})^{\mathrm{an}}$ the point given by coordinates $[\kappa_{0}:\dots:\kappa_{d}]$ .

00LX

Claim 5.2. For any $i\in\{0,\dots,d\}$ , one has

\lVert T_{i}\rVert_{\phi}=r_{i}=\lvert T_{i}\rvert_{\phi}(x(\boldsymbol{r})).

In other words, the maximum of the function $\lvert T_{i}\rvert_{\phi}(x)$ on $(\mathbb{P}^{d}_{k})^{\mathrm{an}}$ is $r_{i}$ , and the maximum values of these $d+1$ functions can be attained at the same point $x(\boldsymbol{r})$ .

00LY

Proof. By the orthogonality of the basis $\{T_{i}\}_{i\in\{0,\dots,d\}}$ , we can compute

\begin{split}\lvert T_{i}\rvert_{\phi}(x(\boldsymbol{r}))&=\inf_{\begin{subarray}{c}(\sum_{m\in\{0,\dots,d\}}f_{m}\cdot T_{m})(x(\boldsymbol{r}))=(T_{i})(x(\boldsymbol{r}))\\ (f_{0},\dots,f_{d})\in k^{d+1}\end{subarray}}\Big\lVert\sum_{m\in\{0,\dots,d\}}f_{m}\cdot T_{m}\Big\rVert_{\phi}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}\kappa_{m}=\kappa_{i}}\max_{m\in\{0,\dots,d\}}\Big\{\lVert f_{m}\cdot T_{m}\rVert_{\phi}\Big\}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}\kappa_{m}=\kappa_{i}}\max_{m\in\{0,\dots,d\}}\Big\{\lvert f_{m}\rvert\lvert\kappa_{m}\rvert\Big\}\\ &=\inf\limits_{\sum_{m\in\{0,\dots,d\}}f_{m}(\kappa_{m}/\kappa_{i})=1}\max_{m\in\{0,\dots,d\}}\Big\{\lvert f_{m}\rvert\lvert\kappa_{m}/\kappa_{i}\rvert\cdot\lvert\kappa_{i}\rvert\Big\}\\ &=\lvert\kappa_{i}\rvert=r_{i}.\end{split}

The last equality is obtained by Lemma 3.13. ∎

By this Claim, for any multi-index $J$ , the function $\lvert\boldsymbol{T}^{J}\rvert_{\phi}(x)$ can attain its maximum value $\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}$ at the point $x(\boldsymbol{r})\in(\mathbb{P}_{k}^{d})^{an}$ as the product of maximum of factors of the monomial. By definition,

\lVert\boldsymbol{T}^{J}\rVert_{|J|\phi}=\sup_{x\in(\mathbb{P}^{d})^{an}}\lvert\boldsymbol{T}^{J}\rvert_{\phi}(x)=\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}=\prod_{i\in\{0,\dots,d\}}\lVert T_{i}\rVert_{\phi}^{j_{i}}.

∎

Second, one calculates the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ on any (homogeneous) combination of monomials. For a general metric, one needs a $\mathbb{Q}$ -independence assumption to gain finer orthogonality.

00LZ

Proposition 5.3. Assume that $\{\alpha(\lVert T_{i}\rVert_{\phi})\}_{i\in\{0,\dots,d\}}$ are $\mathbb{Q}$ -independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$ . Let $S\subseteq\mathbb{N}^{d+1}$ be a finite set of multi-indices, then for any $J\in S$ any $f_{J}\in k$ , one has

\Big\vvvert\sum_{\begin{subarray}{c}J\in S\end{subarray}}f_{J}\cdot\boldsymbol{T}^{J}\Big\vvvert=\sup_{\begin{subarray}{c}J\in S\end{subarray}}\ \lVert f_{J}\cdot\boldsymbol{T}^{J}\rVert_{|J|\phi}.

In other words, the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ on $V_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1))$ is a Gauss norm on $k[T_{0},\dots,T_{d}]$ of multi-radius $\boldsymbol{r}$ . The Banach $k$ -algebra $\widehat{V}_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1),\phi)$ is an affinoid algebra.

00M0

Proof. By the $\mathbb{Q}$ -independence assumption and Proposition 5.1, for any two distinct multi-index $J$ and $J^{\prime}$ , and any two non-zero coefficients $f_{J}$ and $f_{J^{\prime}}$ in $k$ , we have

\lVert f_{J}\cdot\boldsymbol{T}^{J}\rVert_{|J|\phi}=\lvert f_{J}\rvert\cdot\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}\neq\lvert f_{J^{\prime}}\rvert\cdot\prod_{i\in\{0,\dots,d\}}r_{i}^{j^{\prime}_{i}}=\lVert f_{J^{\prime}}\cdot\boldsymbol{T}^{J^{\prime}}\rVert_{|J^{\prime}|\phi}.

By Lemma 2.13, the elements $\{\boldsymbol{T}^{J}\}_{J\in S}$ form an orthogonal basis for the normed vector space $(\bigoplus_{J\in S}k\cdot\boldsymbol{T}^{J},\vvvert\mathord{\cdot}\vvvert_{\phi})$ . So the equality in the conclusion holds. ∎

00M1

Corollary 5.4. With the same assumptions as above, the envelop metric $\mathcal{P}(\phi)$ is a Fubini-Study metric induced by $\lVert\mathord{\cdot}\rVert_{\phi}$ , and is continuous.

For a Fubini-Study metric, one does not need the $\mathbb{Q}$ -independence. For any $\boldsymbol{\delta}=(\delta_{0},\dots,\delta_{d})\in\mathbb{R}^{d+1}$ , one constructs a perturbed metric $\phi(\boldsymbol{\delta})$ as follows. Let $\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})}$ be the norm on $V_{1}(\mathscr{O}(1))$ such that $\{T_{i}\}_{i\in\{1,\dots,d\}}$ is an orthogonal basis with new norms

\forall i\in\{0,\dots,d\},\quad\lVert T_{i}\rVert_{\phi}(\boldsymbol{\delta})=\mathrm{e}^{\delta_{i}}\lVert T_{i}\rVert_{\phi}.

Let $\phi(\boldsymbol{\delta})$ be the metric $\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})})$ on $\mathscr{O}(1)$ . Let $\lvert\boldsymbol{\delta}\rvert$ denote the number $\max_{i\in\{0,\dots,d\}}\lvert\delta_{i}\rvert\in\mathbb{R}_{+}$ .

00M2

Lemma 5.5. Assume that $\phi$ is a Fubini-Study metric. For any $\epsilon>0$ , there exists $\boldsymbol{\delta}\in\mathbb{R}^{d+1}$ with $|\boldsymbol{\delta}|\leq\epsilon$ such that

\forall n\in\mathbb{N},\quad\dist(\lVert\mathord{\cdot}\rVert_{n\phi},\lVert\mathord{\cdot}\rVert_{n\phi(\boldsymbol{\delta})})\leq n\epsilon.

00M3

Proof. Choose an arbitrary $\boldsymbol{\delta}$ with $|\boldsymbol{\delta}|\leq\epsilon$ . Then

\dist(\lVert\mathord{\cdot}\rVert_{\phi},\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})})\leq\epsilon.

By Proposition 3.12, we have

\dist(\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi}),\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi(\boldsymbol{\delta})}))=\dist(\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi}),\phi(\boldsymbol{\delta}))\leq\epsilon,

by the assumption and Proposition 3.11,

\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{\phi})=\phi,

so the conlusion holds. ∎

00M4

Proposition 5.6. Assume that $\phi$ is a Fubini-Study metric. The conclusion of Proposition 5.3 holds without the assumption of $\mathbb{Q}$ -independence of $\{\alpha(\lVert T_{i}\rVert_{1})\}_{i\in\{0,\dots,d\}}$ .

00M5

Proof. Since $\lvert\mathord{\cdot}\rvert_{k}$ is discrete, for any $\epsilon>0$ , there exists $\boldsymbol{\delta}$ with $|\boldsymbol{\delta}|\leq\epsilon$ such that the elements $\{\alpha(\lVert T_{i}\rVert_{\phi(\boldsymbol{\delta})})\}_{i\in\{0,\dots,d\}}$ are $\mathbb{Q}$ -independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$ . By Proposition 5.5, for any $n\in\mathbb{N}$ and any $s_{n}=\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\in V_{n}(\mathscr{O}(1))$ ,

\mathrm{e}^{-n\epsilon}\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi(\boldsymbol{\delta})}\leq\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi}\leq\mathrm{e}^{n\epsilon}\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi(\boldsymbol{\delta})}.

By Proposition 5.3, one has

\max_{\lvert J\rvert=n}\Big\{\mathrm{e}^{-n\epsilon}\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}(\mathrm{e}^{\delta_{i}}r_{i})^{j_{i}}\Big\}\leq\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi}\leq\max_{\lvert J\rvert=n}\Big\{\mathrm{e}^{n\epsilon}\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}(\mathrm{e}^{\delta_{i}}r_{i})^{j_{i}}\Big\}.

Fix $n$ and let $\epsilon\to 0$ , one gets

\Big\lVert\sum_{\lvert J\rvert=n}f_{J}\cdot\boldsymbol{T}^{J}\Big\rVert_{n\phi}=\max_{\lvert J\rvert=n}\Big\{\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}\Big\}.

So one has

\Big\vvvert\sum_{\lvert J\rvert<\infty}f_{J}\cdot\boldsymbol{T}^{J}\Big\vvvert_{n\phi}=\sup_{n\in\mathbb{N}}\max_{\lvert J\rvert=n}\Big\{\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}\Big\}=\max_{\lvert J\rvert<\infty}\Big\{\lvert f_{J}\rvert\prod_{i\in\{0,\dots,d\}}r_{i}^{j_{i}}\Big\}.

Hence $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is a Gauss norm of multi-radius $\boldsymbol{r}$ on $V_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1))$ . ∎

5.1.1. Case for (ℙd,𝒪⁡(1))(\mathbb{P}^{d},\mathscr{O}(1))

5.1.1. Case for $(\mathbb{P}^{d},\mathscr{O}(1))$