2. Reminders on ultrametric functional analysis [00MG]

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2. Reminders on ultrametric functional analysis

In this section, one recalls some facts about functional analysis concerning normed vector spaces and normed algebras over a complete non-Archimedean valued field, following [Ber], [BGR], [FvdP] and [Tem15]. Besides the well-known ones, results in §2.1.2, §2.2.4, §2.3.3 and §2.4 are most relevant to our construction.

Throughout the section, one fixes a field $k$ equipped with a non-Archimedean and non-trivial absolute value $\lvert\mathord{\cdot}\rvert$ and we assume that $k$ equipped with the topology defined by the absolute value is complete. Denote by $k^{\circ}$ the valuation ring of $(k,\lvert\mathord{\cdot}\rvert)$ , by $k^{\circ\circ}$ the maximal ideal of $k^{\circ}$ , and by $\widetilde{k}$ the residual field $k^{\circ}/k^{\circ\circ}$ . Denote by $H(k,\lvert\mathord{\cdot}\rvert)$ the $\mathbb{Q}$ -vector subspace of $\mathbb{R}$ generated by the set of numbers $\log\lvert k^{\times}\rvert$ , and by $\alpha$ the quotient map of $\mathbb{Q}$ -vector spaces $\mathbb{R}\to\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$ . One says that $n$ real numbers $\{p_{1},\dots,p_{n}\}$ are $\mathbb{Q}$ -independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$ if the vectors $\{\alpha(p_{1}),\dots,\alpha(p_{n})\}$ are $\mathbb{Q}$ -linearly independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$ . Unless specified, all $k$ -algebras are supposed to be commutative unitary (with $0\neq 1$ ) and by convention all homomorphism of $k$ -algebras are supposed to preserve the units.

2.1. Seminormed vector spaces

2.1.1. Basic constructions

Let $V$ be a vector space over $k$ . By seminorm on $V$ , we refer to a map $\lVert\mathord{\cdot}\rVert:V\rightarrow\mathbb{R}_{\geq 0}$ such that $\lVert ax\rVert=\lvert a\rvert\cdot\lVert x\rVert$ for any $(a,x)\in k\times V$ and that $\lVert x+y\rVert\leqslant\lVert x\rVert+\lVert y\rVert$ for any $(x,y)\in V\times V$ . The couple $(V,\lVert\mathord{\cdot}\rVert)$ is called a seminormed vector space over $k$ . If in addition $\lVert\mathord{\cdot}\rVert$ takes positive values on $V\setminus 0$ , we say that $\lVert\mathord{\cdot}\rVert$ is a norm on $V$ and that $(V,\lVert\mathord{\cdot}\rVert)$ is a normed vector space. Denote by $\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ the inverse image of $\{0\}$ by $\lVert\mathord{\cdot}\rVert$ . It can be shown that $\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ is a closed vector subspace of $V$ , called the null space of $\lVert\mathord{\cdot}\rVert$ . Note that there exists a unique norm on $V/\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ , the composition of which with the projection map $V\rightarrow V/\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ identifies with the seminorm $\lVert\mathord{\cdot}\rVert$ . Call this norm the induced norm of $\lVert\mathord{\cdot}\rVert$ .

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a seminormed vector space over $k$ . If the strong triangle inequality holds for the seminorm $\lVert\mathord{\cdot}\rVert$ , namely $\lVert x+y\rVert\leqslant\max(\lVert x\rVert,\lVert y\rVert)$ for any $(x,y)\in V\times V$ , we say that the seminorm is ultrametric. Note that if a seminorm is ultra-metric, then the inequality becomes an equality whenever $\lVert x\rVert\neq\lVert y\rVert$ .

We say that a seminormed (resp. normed) vector space $(V,\lVert\mathord{\cdot}\rVert)$ is complete, or $\lVert\mathord{\cdot}\rVert$ is a complete seminorm (resp. complete norm) on $V$ , if any Cauchy sequence in $V$ with respect to the seminorm $\lVert\mathord{\cdot}\rVert$ admits a limit. A complete normed vector space over $k$ is called a Banach space over $k$ . Any finite-dimensional normed space $(V,\lVert\mathord{\cdot}\rVert)$ is complete. ([Bou, 1.2.3 Theorem 2])

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ be a seminormed vector space over $k$ . Let $\widetilde{V}_{c}$ be the vector space of all Cauchy sequences in $V$ with respect to $\lVert\mathord{\cdot}\rVert_{V}$ . We define a seminorm $\lVert\mathord{\cdot}\rVert_{c}$ on $\widetilde{V}_{c}$ which sends any Cauchy sequence $\{v_{i}\}_{i\in\mathbb{N}}$ to $\lim_{i\rightarrow+\infty}\lVert v_{i}\rVert_{V}$ . Denote by $V_{c}$ the quotient vector space $V_{c}/\mathfrak{n}(\lVert\mathord{\cdot}\rVert_{c})$ . Then the vector space $V_{c}$ equipped with the norm induced by $\lVert\mathord{\cdot}\rVert_{c}$ forms a Banach space over $k$ , called the separated completion of $(V,\lVert\mathord{\cdot}\rVert)$ . Tautologically it can be shown that this Banach space is canonically isomorphic to the completion of $V/\mathfrak{n}(\lVert\mathord{\cdot}\rVert)$ equipped with the quotient norm induced by the seminorm $\lVert\mathord{\cdot}\rVert$ .

Definition 2.1.

Let $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ be seminorms on $V$ . We say that $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ are equivalent if there exist two constants $C_{1}>0$ and $C_{2}>0$ such that $C_{1}\lVert\mathord{\cdot}\rVert_{1}\leq\lVert\mathord{\cdot}\rVert_{2}\leq C_{2}\lVert\mathord{\cdot}\rVert_{1}$ . Note that this condition holds if and only if the seminorms $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ induce the same topology on the vector space $V$ ([Bou, Corollaire I.3.3.1])(note that the absolute value $\lvert\mathord{\cdot}\rvert$ is supposed to be non-trivial).

Definition 2.2.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a seminormed vector space over $k$ . If $W$ is a vector subspace of $V$ , then map $(x\in W)\mapsto\lVert x\rVert$ defines a seminorm on $W$ , called the restriction of $\lVert\mathord{\cdot}\rVert$ on $W$ . If $Q$ is a quotient vector space of $V$ and $\pi:V\rightarrow Q$ is the quotient map, then the map $(q\in Q)\mapsto\inf_{x\in\pi^{-1}(\{q\})}\lVert x\rVert$ defines a seminorm on $Q$ , called the quotient of $\lVert\mathord{\cdot}\rVert$ on $Q$ .

Definition 2.3.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be seminormed vector spaces over $k$ , and $f:V\rightarrow W$ be a $k$ -linear map. We say that $f$ is bounded if there exists a constant $C>0$ such that $\lVert f(x)\rVert_{W}\leq C\lVert x\rVert_{V}$ for any $x\in V$ . Note that this condition holds if and only if $f$ is continuous with respect to the topologies on $V$ and $W$ induced by the seminorms $\lVert\mathord{\cdot}\rVert_{V}$ and $\lVert\mathord{\cdot}\rVert_{W}$ respectively. We say that $f$ is admissible if it is bounded and if on the image of $f$ , the quotient seminorm of $\lVert\mathord{\cdot}\rVert_{V}$ and the restriction of $\lVert\mathord{\cdot}\rVert_{W}$ are equivalent.

We recall below several fundamental results in functional analysis and refer to [Bou, Theorem 1.3.3.1, Corollary 1.3.3.1, 1.3.3.2, 1.3.3.5] for more details.

Theorem 2.4.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be Banach spaces over $k$ , and $f:V\rightarrow W$ be a $k$ -linear map.

(1)

The $k$ -linear map $f$ is bounded if and only if its graph in $V\times W$ is closed under the product topology.
(2)

Assume that $f$ is bounded and surjective, then $f$ is an open map. In particular, the quotient norm of $\lVert\mathord{\cdot}\rVert_{V}$ on $W$ is equivalent to $\lVert\mathord{\cdot}\rVert_{W}$ .
(3)

Assume that $f$ is bounded and injective, then $f(V)$ is closed in $W$ .

Theorem 2.5.

Let $V$ be a vector space over $k$ and $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ be complete norms on $V$ . If there exists $C>0$ such that $\lVert\mathord{\cdot}\rVert_{2}\leq C\lVert\mathord{\cdot}\rVert_{1}$ , then the norms $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ are equivalent.

Using this norm equivalence theorem for Banach spaces over $k$ , we have immediately the following

Corollary 2.6.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be Banach spaces over $k$ , and $f:V\rightarrow W$ be a bounded $k$ -linear map with closed image. Then $f$ is admissible.

Definition 2.7.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional normed vector space. The dual norm of $\lVert\mathord{\cdot}\rVert^{\vee}$ on the dual vector space $V^{\vee}$ is defined by

\forall\ell\in V^{\vee},\ \lVert\ell\rVert^{\vee}:=\sup_{v\in V\setminus\{0\}}\frac{\lvert\ell(v)\rvert}{\lVert v\rVert}.

Remark 2.8.

The norm $\lVert\mathord{\cdot}\rVert^{\vee}$ is ultrametric, and $\lVert\mathord{\cdot}\rVert^{\vee\vee}=\lVert\mathord{\cdot}\rVert$ if and only if $\lVert\mathord{\cdot}\rVert$ is ultrametric. ([CMor18, Section 2.2.3])

Definition 2.9.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a normed vector space. Let $(k^{\prime},\lvert\mathord{\cdot}\rvert^{\prime})$ be a complete valued field extension of $(k,\lvert\mathord{\cdot}\rvert)$ . Set $V_{k^{\prime}}$ to be $V\otimes_{k}k^{\prime}$ , which can be identified with $\mathrm{Hom}_{k}(\mathrm{Hom}_{k}(V,k),k^{\prime})$ . The norm

\forall v^{\prime}\in V_{k^{\prime}},\ \lVert v^{\prime}\rVert_{k^{\prime}}:=\sup\Big\{\frac{\lvert(\ell\otimes 1)(v^{\prime})\rvert^{\prime}}{\lVert\ell\rVert^{\vee}},\ \ell\in V^{\vee}\Big\}

defined via this identification is called the scalar extension of $\lVert\mathord{\cdot}\rVert$ .

Remark 2.10.

If $\lVert\mathord{\cdot}\rVert$ is ultrametric, then $\lVert\mathord{\cdot}\rVert_{k^{\prime}}$ is the largest ultrametric norm on $V_{k^{\prime}}$ extending $\lVert\mathord{\cdot}\rVert$ . ([CMor18, Definition 2.4])

Lemma 2.11.

Let $f:V\rightarrow W$ be a surjective $k$ -linear map of finite-dimensional vector spaces, with $\mathrm{dim}_{k}W=1$ . Let $\lVert\mathord{\cdot}\rVert_{V}$ be a norm on $V$ and let $\lVert\mathord{\cdot}\rVert_{W}$ be its quotient norm for $f$ . Then the norm $\lVert\mathord{\cdot}\rVert_{W,k^{\prime}}$ identifies with the quotient norm of $\lVert\mathord{\cdot}\rVert_{V,k^{\prime}}$ induced by the surjective $k^{\prime}$ -linear map $f\otimes\mathrm{id}_{k^{\prime}}:V_{k^{\prime}}\rightarrow W_{k^{\prime}}$ . ([CMor18, Lemma 2.5])

2.1.2. Orthogonal basis

Definition 2.12.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional normed vector space over $k$ . A basis $\{e_{i}\}_{i\in\{1,\dots,n\}}$ of $V$ is called orthogonal (with respect to $\lVert\mathord{\cdot}\rVert$ ) if

\forall(c_{1},\dots,c_{r})\in k^{n},\quad\Big\|\sum_{i\in\{1,\dots,n\}}c_{i}e_{i}\Big\|=\max_{i\in\{1,\dots,n\}}\lVert c_{i}e_{i}\rVert

Moreover, it is said to be orthonormal if in addition $\lVert e_{i}\rVert=1$ for all $i\in\{1,\dots,n\}$ .

Lemma 2.13.

Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional ultrametrically normed vector space over $k$ . If $\{v_{i}\}_{i\in\{1,\dots,n\}}$ is a finite set of elements of $V$ such that $\{\lVert v_{i}\rVert\}_{i\in\{1,\dots,n\}}$ are disctinct in $\mathbb{R}_{+}$ . Then $\lVert\sum_{i\in\{1,\dots,n\}}v_{i}\rVert=\max_{i\in\{1,\dots,n\}}\lVert v_{i}\rVert$ .

Proof.

If $n=2$ , this is clear from the ultra-metric inequality. For general $n$ an induction argument shows the equality. ∎

Corollary 2.14.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ be a finite-dimensional ultrametrically normed vector space over $k$ . Suppose that $(k,|\cdot|)$ is discretely valued. If $\{e_{i}\}_{i\in\{1,\dots,n\}}$ is a basis of $V$ such that $\{\log\lVert e_{i}\rVert\}_{i\in\{1,\dots,n\}}$ are $\mathbb{Q}$ -independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$ , then $\{e_{i}\}_{i\in\{1,\dots,n\}}$ is an orthogonal basis.

Proof.

For any $f=(f_{1},\dots,f_{n})\in(k^{\times})^{n}$ , the numbers $\{\lvert f_{i}\rvert\lVert e_{i}\rVert\}_{i\in\{1,\dots,r\}}$ are distinct, otherwise there exist $i,j\in\{1,\dots,n\},i\neq j$ such that

\log\lVert e_{i}\rVert-\log\lVert e_{j}\rVert=\log\Big|\frac{f_{i}}{f_{j}}\Big|\in\log\lvert k^{\times}\rvert

which contradicts the assumption of $\mathbb{Q}$ -independence. Hence

\Big\|\sum_{i\in\{1,\dots,n\}}f_{i}e_{i}\Big\|=\max_{0\leq i\leq n}\lVert f_{i}e_{i}\rVert

by Lemma 2.13. ∎

Proposition 2.15.

Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ be a finite-dimensional ultrametrically normed vector space over $k$ . Suppose that $(k,\lvert\mathord{\cdot}\rvert)$ is discretely valued. Then there exists an orthogonal basis for $(V,\lVert\mathord{\cdot}\rVert)$ . ([BMPS, Proposition 2.5])

2.2. Banach algebra

2.2.1. Basic constructions

Definition 2.16.

Let $A$ be a $k$ -algebra (the unit of which is denoted by $\mathbf{1}$ ) and $\lVert\mathord{\cdot}\rVert$ be a seminorm on $A$ (viewed as a vector space over $k$ ).

(1)

The seminorm $\lVert\mathord{\cdot}\rVert$ is said to be sub-multiplicative if for any $(a,b)\in A\times A$ one has $\lVert ab\rVert\leq\lVert a\rVert\cdot\lVert b\rVert$ .
(2)

The seminorm $\lVert\mathord{\cdot}\rVert$ is called power-multiplicative if $\lVert a^{n}\rVert=\lVert a\rVert^{n}$ for any $a\in A$ and any $n\in\mathbb{N}\setminus\{0\}$ .
(3)

The seminorm $\lVert\mathord{\cdot}\rVert$ is called multiplicative if $\lVert ab\rVert=\lVert a\rVert\cdot\lVert b\rVert$ for any $(a,b)\in A^{2}$ .

A $k$ -algebra seminorm (resp. $k$ -algebra norm) on $A$ is defined to be a sub-multiplicative seminorm (resp. sub-multiplicative norm) $\lVert\mathord{\cdot}\rVert$ on $A$ such that $\lVert\mathbf{1}\rVert=1$ . We denote by $\vvvert\mathord{\cdot}\vvvert$ an algebra seminorm. Any $k$ -algebra equipped with a complete $k$ -algebra norm is called a Banach $k$ -algebra.

We use calligraphic letters to denote Banach algebras and Banach modules (defined below) and use the corresponding capital letters to denote the underlying $k$ -algebra or the underlying module of a $k$ -algebra. For example, a Banach $k$ -algebra $(A,\vvvert\mathord{\cdot}\vvvert)$ is denoted by $\mathcal{A}$ . If $A^{\prime}$ is a sub- $k$ -algebra of $A$ , then the restriction of $\vvvert\mathord{\cdot}\vvvert$ on $A^{\prime}$ is a $k$ -algebra norm. If this norm is complete, we say that $\mathcal{A}^{\prime}$ ( $A^{\prime}$ equipped with the restricted norm) is a Banach $k$ -sub-algebra of $\mathcal{A}$ . Similarly, if $Q$ is a quotient $k$ -algebra of $A$ , then the quotient of the norm $\vvvert\mathord{\cdot}\vvvert$ on $Q$ is a sub-multiplicative seminorm. If it is a complete norm, we say that $\mathcal{Q}$ ( $Q$ equipped with the quotient norm) is a Banach quotient $k$ -algebra of $\mathcal{A}$ .

Example 2.17.

Let $\mathcal{A}$ be a Banach $k$ -algebra. The Tate $k$ -Banach algebra over $\mathcal{A}$ of multiradius $\boldsymbol{r}=(r_{1},\dots,r_{n})\in(\mathbb{R}_{+})^{N}$ is the algebra over $k$

\Big\{\sum_{J\in\mathbb{N}^{n}}a_{J}\boldsymbol{T}^{J},\text{ }a_{J}\in A\text{ and }\lim_{|J|\to\infty}\vvvert a_{J}\vvvert\cdot\boldsymbol{r}^{J}=0\Big\}

(for $J=(j_{1},\dots,j_{n})\in\mathbb{N}^{n}$ , we denote $\prod_{i\in\{1,\dots,n\}}T_{i}^{j_{i}}$ by $\boldsymbol{T}^{J}$ and $\prod_{i\in\{1,\dots,n\}}r_{i}^{j_{i}}$ by $\boldsymbol{r}^{J}$ ) with a complete $k$ -algebra norm defined by

\Big\vvvert\sum_{J\in\mathbb{N}^{n}}a_{J}\boldsymbol{T}^{J}\Big\vvvert_{\mathcal{T}_{\mathcal{A}}(\boldsymbol{r})}:=\sup_{J}\vvvert a_{J}\vvvert\cdot\boldsymbol{r}^{J}

This Banach algebra is denoted by $\mathcal{A}\{r_{1}^{-1}T_{1},\dots,r_{n}^{-1}T_{n}\}$ , and is called an $\mathcal{A}$ -Tate algebra of multiradius $\boldsymbol{r}$ .

Definition 2.18.

Let $\mathcal{A}_{1},\mathcal{A}_{2}$ be two Banach $k$ -algebras, and $\phi:A_{1}\to A_{2}$ be a homomorphism of $k$ -algebras. We say that $\phi$ is a homomorphism of Banach $k$ -algebras if it is bounded as a $k$ -linear map. A homomorphism of Banach $k$ -algebra $\phi$ is often denoted by $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ . A homomorphism of Banach $k$ -algebra $\phi$ is called an isomorphism of Banach $k$ -algebras if there exists a homomorphism of Banach $k$ -algebras $\psi:\mathcal{A}_{2}\to\mathcal{A}_{1}$ such that $\phi\circ\psi=\mathrm{Id}_{\mathcal{A}_{2}}$ and $\psi\circ\phi=\mathrm{Id}_{\mathcal{A}_{1}}$ .

2.2.2. Spectrum

Let $\mathcal{A}=(A,\vvvert\mathord{\cdot}\vvvert)$ be a Banach $k$ -algebra. Let $\vvvert\mathord{\cdot}\vvvert^{\prime}$ be a $k$ -algebra seminorm on $A$ . One says that $\vvvert\mathord{\cdot}\vvvert^{\prime}$ is bounded (with respect to $\mathcal{A}$ ) if there exists $C>0$ such that $\vvvert\mathord{\cdot}\vvvert^{\prime}\leq C\vvvert\mathord{\cdot}\vvvert$ . Its null-space is a closed ideal $I$ of $A$ ; the quotient $k$ -algebra norm of $\vvvert\mathord{\cdot}\vvvert^{\prime}$ on the quotient $k$ -algebra $A/I$ is bounded with respect to the quotient $k$ -algebra norm of $\vvvert\mathord{\cdot}\vvvert$ . ([Ber, Remark 1.2.2.i])

Definition 2.19.

Let $\mathcal{A}$ be a $k$ -Banach algebra. The Berkovich spectrum $\mathfrak{M}(\mathcal{A})$ is the following topological space: the points, denoted by $z$ , are bounded multiplicative $k$ -algebra seminorms $\vvvert\mathord{\cdot}\vvvert_{z}$ on $\mathcal{A}$ , and the topology is the weakest topology on this set of points, for which all $\mathbb{R}_{\geq 0}$ -valued functions of the form $z\mapsto\vvvert f\vvvert_{z}$ are continuous for any $f\in A$ . This topology is called the canonical topology. For any subset $V$ of $\mathfrak{M}(\mathcal{A})$ , we denote by $\text{Int}^{\mathrm{top}}(V)$ the topological interior of $V$ . This topological interior is to be compared with the notion of interior of an affinoid subdomain in an affinoid domain (see [Ber, Definition 2.5.7]), which we do not use in this article.

Remark 2.20.

A basis for the canonical topology constituting of open sets is given by basic open sets, which are sets of the form

U(f;p,q):=\{z\in\mathfrak{M}(\mathcal{A}):p<\lvert f\rvert_{z}<q\}

indexed by $(p,q)\in\mathbb{R}^{2}$ and $f\in\mathcal{A}$ . A general open set is a union of finite intersections of basic open sets.

Proposition 2.21.

Let $\mathcal{A}$ be a $k$ -Banach algebra. Then $\mathfrak{M}(\mathcal{A})$ is a non-empty compact Hausdorff topological space. ([Ber, Theorem 1.2.1])

For any point $z\in\mathfrak{M}(A)$ , let $\mathfrak{p}_{z}$ be the closed ideal $\mathfrak{n}(\vvvert\mathord{\cdot}\vvvert_{z})$ of $A$ which is a prime ideal, and $f(z)$ be the image of $f$ in the quotient $k$ -algebra $A/\mathfrak{p}_{z}$ . The residual field at $z$ is defined to be the fraction field of $A/\mathfrak{p}_{z}$ , denoted by $\kappa(z)$ , it is equipped with a quotient norm $\lvert\mathord{\cdot}\rvert_{z}$ of $\vvvert\mathord{\cdot}\vvvert_{z}$ , which becomes an absolute value on $\kappa(x)$ extending $\lvert\mathord{\cdot}\rvert$ on $k$ . The completed residual field at $z$ is defined to be the completion of $\lvert\mathord{\cdot}\rvert_{z}$ with respect to this quotient norm $\lvert\mathord{\cdot}\rvert_{z}$ , denoted as $\widehat{\kappa}(z)$ . The canonical homomorphism of $k$ -algebra from $A$ to $(\widehat{\kappa}(z),\lvert\mathord{\cdot}\rvert_{z})$ is denoted by $\chi_{z}$ . It is a homomorphism of Banach $k$ -algebras.

Definition 2.22.

Let $\mathcal{A}$ be a Banach $k$ -algebra. A character $\chi$ of $\mathcal{A}$ is a homomorphism of Banach $k$ -algebra from $\mathcal{A}$ to some complete valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ of $(k,\lvert\mathord{\cdot}\rvert_{k})$ . Two characters $\chi_{1}:\mathcal{A}\to(K_{1},\lvert\mathord{\cdot}\rvert_{K_{1}})$ and $\chi_{2}:\mathcal{A}\to(K_{2},\lvert\mathord{\cdot}\rvert_{K_{2}})$ are said to be equivalent if there exist a character $\chi:\mathcal{A}\to(K,\lvert\mathord{\cdot}\rvert_{K})$ and valued field extensions $\iota_{1}:K\to K_{1}$ and $\iota_{2}:K\to K_{2}$ which preserve norms such that $\chi=i_{1}\circ\chi_{1}=i_{2}\circ\chi_{2}$ . Let $[\chi]$ be the equivalence class of $\chi$ .

Lemma 2.23.

The set of points of $\mathfrak{M}(\mathcal{A})$ is in canonical bijection with the set of equivalence classes of characters on $\mathcal{A}$ . This bijection sends $z\in\mathfrak{M}(\mathcal{A})$ to $[\chi_{z}]$ . ([Ber, Remark 1.2.2.ii])

Definition 2.24.

The Gelfand transform of $\mathcal{A}$ is the homomorphism of Banach $k$ -algebras

\widehat{}:A\to\prod_{z\in\mathfrak{M}(\mathcal{A})}\hat{\kappa}(z),\quad f\mapsto\widehat{f}=(f(z))_{z\in\mathfrak{M}(\mathcal{A})}

Proposition 2.25.

An element $f\in\mathcal{A}$ is invertible if and only if $f(z)\neq 0$ for any $z\in\mathfrak{M}(\mathcal{A})$ . ([Ber, Corollary 1.2.4])

2.2.3. Continuous map

Proposition 2.26.

Let $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a homomorphism of Banach $k$ -algebras. It induces a continuous map $\phi^{\star}:\mathfrak{M}(\mathcal{A}_{2})\to\mathfrak{M}(\mathcal{A}_{1})$ by sending an equivalent class of characters $[\chi]$ of $\mathcal{A}_{2}$ to the class of characters $[\chi\circ\psi]$ of $\mathcal{A}_{1}$ . ([Ber, Remark 1.2.2 (iii)])

Lemma 2.27.

If $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ is a homomorphism of Banach $k$ -algebras with dense image, then $\phi^{\star}$ is an injective map whose image is closed.

Proof.

The map $\phi^{\star}$ is injective since for any two characters $\chi_{1},\chi_{2}:\mathcal{A}_{2}\to K$ , if $\chi_{1}\circ\phi=\chi_{2}\circ\phi$ , then the restriction of $\chi_{1}$ and $\chi_{2}$ on the image of $\phi$ are equal, hence the two characters are equal by the density of image.

Let $(z,\lvert\mathord{\cdot}\rvert_{z})\in\mathfrak{M}(\mathcal{A}_{1})$ which is not in the image of $\phi^{\star}$ , then $\ker(\phi)\nsubseteq\mathfrak{p}_{z}$ : otherwise the character $\mathcal{A}_{1}/\ker(\phi)\to\hat{\kappa}(z)$ extends to a character $\mathcal{A}_{2}\to\hat{\kappa}(z)$ by the density of image of $\phi$ . Now there exists $f\in\ker(\phi)\setminus\mathfrak{p}_{z}$ , so $|f|_{z}\neq 0$ . For small enough $\epsilon>0$ , the basic open set $U(f;|f|_{z}-\epsilon,|f|_{z}+\epsilon)\subset\mathfrak{M}(\mathcal{A}_{1})$ is a neighbourhood of $(z,\lvert\mathord{\cdot}\rvert_{z})$ which is not contained in the image of $\phi^{\star}$ . So the image of $\phi^{\star}$ is a closed subset in $\mathfrak{M}(\mathcal{A}_{1})$ . ∎

2.2.4. Spectral seminorm

Definition 2.28.

The spectral algebra seminorm $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ of an algebra seminorm $\vvvert\mathord{\cdot}\vvvert$ on a $k$ -algebra $A$ is the one defined by

\forall f\in\mathcal{A},\quad\vvvert f\vvvert_{\mathrm{sp}}:=\lim_{\begin{subarray}{c}n\to\infty\end{subarray}}\vvvert f^{n}\vvvert^{\frac{1}{n}}.

Note that the triangle inequality for $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ follows from sub-multiplicativity of $\vvvert\mathord{\cdot}\vvvert$ . In general, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is only a seminorm even if $\vvvert\mathord{\cdot}\vvvert$ is a norm.

Remark 2.29.

The existence of limit is guaranteed by the (multiplicative) Fekete lemma for the sub-multiplicative sequence $\{\vvvert f^{n}\vvvert\}_{n\in\mathbb{N}}$ . The spectral seminorm is sub-multiplicative, and is bounded by the original seminorm $\vvvert\mathord{\cdot}\vvvert$ . Moreover, it is power-multiplicative by construction.

Proposition 2.30.

Let $\mathcal{A}$ be a $k$ -Banach algebra. For any $f\in A$ , one has ([Ber, Theorem 1.3.1])

\vvvert f\vvvert_{\mathrm{sp}}=\max_{z\in\mathfrak{M}(A)}|f|_{z}

Definition 2.31.

Let $\mathcal{A}$ be a Banach $k$ -algebra. The radical of $\mathcal{A}$ is the null-space of its spectral seminorm $\mathfrak{n}(\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}})$ . A Banach $k$ -algebra with radical equal to $\{0\}$ is called semi-simple. Elements in the radical are said to be quasi-nilpotent (or topological nilpotent).

Remark 2.32.

The radical of $\mathcal{A}$ contains the nil-radical of $A$ ; in other words, nilpotent elemtents are quasi-nilpotent. If $\mathcal{A}$ is semi-simple, then $A$ is reduced. The converse may not be true.

Let $\mathcal{A}=(A,\vvvert\mathord{\cdot}\vvvert)$ be a $k$ -Banach algebra. The spectral seminorm $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ defines a quotient norm on the quotient $k$ -algebra $\mathcal{A}/\text{rad}(\mathcal{A})$ , still denoted by $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ . The quotient norm $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is bounded by the quotient norm of $\vvvert\mathord{\cdot}\vvvert$ . The uniformization $\mathcal{A}^{u}$ of $\mathcal{A}$ is defined to be the Banach $k$ -algebra of separated completion of $(\mathcal{A}/\text{rad}(\mathcal{A}),\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}})$ . Conversely, if $\vvvert\mathord{\cdot}\vvvert$ is a power-multiplicative Banach algebra norm on $A$ with radical $\{0\}$ , then it is said to be uniform.

Obviously, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is bounded by $\vvvert\mathord{\cdot}\vvvert$ . It is important to note that the converse may not be true in general. In other words, $\vvvert\cdot\vvvert_{\mathrm{sp}}$ may not be complete on $\mathcal{A}/\text{rad}(\mathcal{A})$ . Yet one still has the following statement

Proposition 2.33.

$\mathfrak{M}(\mathcal{A})$ is canonically homeomorphic to $\mathfrak{M}(\mathcal{A}^{u})$ . ([Ber, Corollary 1.3.3, 1.3.4])

2.2.5. Banach module

One can also consider seminorms on modules over Banach algebra. Let $\mathcal{A}$ be a Banach $k$ -algebra. A (semi)normed $\mathcal{A}$ -module is defined to be an $A$ -module $M$ with a (semi)norm $\lVert\mathord{\cdot}\rVert$ such that $(M,\lVert\mathord{\cdot}\rVert)$ is a (semi)normed vector space over $k$ (denoted by $\mathcal{M}$ ), and that the multiplication is bounded, in the sense that there exists $C>0$ such that

\forall a\in\mathcal{A},\ \forall m\in M,\quad\lVert a\cdot m\rVert\leq C\vvvert a\vvvert\cdot\lVert m\rVert

One calls a Banach $\mathcal{A}$ -module a normed $\mathcal{A}$ -module $(M,\lVert\mathord{\cdot}\rVert)$ whose norm is complete.

Let $\mathcal{M}_{1}=(M_{1},\lVert\mathord{\cdot}\rVert_{1})$ , $\mathcal{M}_{2}=(M_{2},\lVert\mathord{\cdot}\rVert_{2})$ be Banach $\mathcal{A}$ -modules and $\phi:M_{1}\to M_{2}$ be a homomorphism of $A$ -modules. It is called bounded if there exists $C>0$ such that $\lVert\phi(m_{1})\rVert_{2}\leq C\lVert m_{1}\rVert_{1}$ for any $m_{1}\in M_{1}$ . In this case $\phi$ is said to be a homomorphism of Banach $\mathcal{A}$ -modules, and is denoted by $\phi:\mathcal{M}_{1}\rightarrow\mathcal{M}_{2}$ . In addition, the homomorphism $\phi$ of Banach $\mathcal{A}$ -modules is called admissible if it is admissible as linear map between normed-vector spaces over $k$ .

Definition 2.34.

Let $\mathcal{M}$ be a Banach $\mathcal{A}$ -module. It is called a Banach finite $\mathcal{A}$ -module if there exists $l\in\mathbb{N}_{+}$ and a surjective homomorphism of Banach $\mathcal{A}$ -modules $\mathcal{A}^{\oplus l}\to\mathcal{M}$ where $\mathcal{A}^{\oplus l}$ is the Banach $\mathcal{A}$ -module corresponding to the $A$ -module $A^{\oplus l}$ equipped with the norm $(a_{1},\dots,a_{l})\mapsto\max\vvvert a_{i}\vvvert$ . (Note that such a homomorphism is necessarily admissible.)

Proposition 2.35.

Let $\mathcal{A}$ be a Banach $k$ -algebra and $\mathcal{M}$ be a Banach $\mathcal{A}$ -module. If $A$ is Noetherian as a $k$ -algebra and $M$ is finitely generated as $A$ -module, then any $\mathcal{A}$ -sub-module of $\mathcal{M}$ is closed, and $\mathcal{M}$ is a Banach finite $\mathcal{A}$ -module. ([FvdP, Lemma 1.2.3]

Definition 2.36.

Let $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a homomorphism between Banach $k$ -algebras. It is called Banach finite if $\mathcal{A}_{2}$ is a Banach finite $\mathcal{A}_{1}$ -module. In this case $\mathcal{A}_{2}$ is called a Banach finite $\mathcal{A}_{1}$ -algebra.

Remark 2.37.

If a $k$ -Banach algebra homomorphism $\phi$ is finite as homomorphism of $k$ -algebra, and $\mathcal{A}_{1}$ is Noetherian, then $\phi$ is automatically Banach finite: there is a surjective $\mathcal{A}_{1}$ -module homomorphism $p:\mathcal{A}_{1}^{\oplus n}\to\mathcal{A}_{2}$ , by Proposition 2.35 $\ker(p)$ is closed. Then $p$ is continuous hence is admissible by Corollary 2.5. So $\mathcal{A}_{2}$ is a Banach finite $\mathcal{A}_{1}$ -module.

2.3. Affinoid algebras

Affinoid algebras is a special kind of $k$ -Banach algebras possessing good finiteness properties. These features allows one to endow a locally ringed space structure on their Berkovich spectra, namely the affinoid spaces. As a consequence, the Banach algebra norm of an affinoid algebra is equivalent to its spectral seminorm whenever the later is actually a norm.

2.3.1. Basic constructions

Affinoid algebras are $k$ -Banach algebras that are quotient algebras of Tate algebras. Among them are strict affinoid algebras which have good finiteness properties such as Noetherianity. Some good properties pass to general affinoid algebra by a technique enlarging the base valued field which makes the affinoid algebra strict.

Definition 2.38.

For a multi-radius $\boldsymbol{r}=(r_{1},\dots,r_{n})\in\mathbb{R}^{n}$ , the algebra

k\{r_{1}^{-1}T_{1},\dots,r_{n}^{-1}T_{n}\}=\{f=\sum_{J\in\mathbb{N}^{n}}^{\infty}a_{J}\boldsymbol{T}^{J}:a_{J}\in k,\lvert a_{J}\rvert\boldsymbol{r}^{J}\to 0\text{ as }|J|\to\infty\}

is called the Tate algebra over $k$ with multi-radius $\boldsymbol{r}$ . Denote it by $\mathcal{T}_{n}(\boldsymbol{r})$ . It is a $k$ -Banach algebra with respect to the Gauss norm of multi-radius $\boldsymbol{r}$ defined by

\vvvert f\vvvert_{\mathcal{T}_{n}(\boldsymbol{r})}=\max_{J}\lvert a_{J}\rvert\boldsymbol{r}^{J}

One can define Tate algebra over other complete ultra-metric valued fields.

Remark 2.39.

This Gauss norm is obviously sub-multiplicative. It is in fact multiplicative by an argument as in the proof of Gauss Lemma.

Definition 2.40.

A $k$ -Banach algebra $\mathcal{A}$ is called an affinoid algebra if there exists an admissible surjective homomorphism from some Tate algebra $k\{\boldsymbol{r}^{-1}\boldsymbol{T}\}$ to $\mathcal{A}$ . The Banach algebra norm on an affinoid algebra $\mathcal{A}$ is called an affinoid algebra norm. If one can take $\boldsymbol{r}$ with $r_{i}=1$ for all $i\in\{1,\dots,n\}$ , then $\mathcal{A}$ is called a strict affinoid algebra. One may define affinoid algebra similarly over other complete ultrametric valued field.

Remark 2.41.

An affinoid algebra norm and the quotient algebra norm of Gauss algebra norm by the defining admissible surjective homomorphism are just equivalent but not necessarily equal.

One can construct new affinoid algebras out of old ones by various algebraic operations.

Example 2.42.

The quotient Banach algebra of an affinoid algebra is an affinoid algebra.

Proposition 2.43.

Let $\mathcal{C}$ be a $k$ -Banach algebra which is finite over an affinoid algebra $\mathcal{A}$ , then $\mathcal{C}$ itself is an affinoid algebra. If $\mathcal{A}$ is strict, then $\mathcal{C}$ is strict.

Proof.

Let $\{c_{i}\}_{i\in\{1,\dots,m\}}\subset\mathcal{C}$ be a finite set of generators of $\mathcal{C}$ over $\mathcal{A}$ , then consider an $\mathcal{A}$ -Tate algebra $\mathcal{A}\{\boldsymbol{r}^{-1}\boldsymbol{T}\}$ where $r_{i}\geq\vvvert c_{i}\vvvert_{\mathcal{C}}$ . There is a surjective $k$ -algebra homomorphism defined by

\gamma:\mathcal{A}\{\boldsymbol{r}^{-1}\boldsymbol{T}\}\to\mathcal{C},\text{ }T_{i}\mapsto c_{i}

which is bounded as there exists $C>0$ such that

\vvvert\gamma(\sum_{J}a_{J}\boldsymbol{T}^{J})\vvvert_{\mathcal{C}}\leq\max_{J\in\mathbb{N}^{m}}\vvvert a_{J}\boldsymbol{c}^{J}\vvvert_{\mathcal{C}}\leq C\max_{J\in\mathbb{N}^{m}}\vvvert a_{J}\vvvert_{\mathcal{A}}\cdot\boldsymbol{r}^{J}

By Corollary 2.5, $\gamma$ is admissible, the norm $\vvvert\mathord{\cdot}\vvvert_{\mathcal{C}}$ is equivalent to the quotient norm of the $\mathcal{A}$ -Tate norm. Hence $\mathcal{C}$ is an affinoid algebra. The strictness is obtained by choosing $r_{i}\in\lvert k^{\times}\rvert$ (see Lemma 2.47). ∎

Proposition 2.44.

Let $\mathcal{B}$ be a Banach $k$ -algebra. Suppose that there exists a finitely generated $k$ -algebra $A$ which is dense in $\mathcal{B}$ , then there exists an affinoid algebra $\mathcal{A}$ in which $A$ is a dense $k$ -sub-algebra and a homomorphism of Banach $k$ -algebras $\mathcal{A}\rightarrow\mathcal{B}$ which extends the identiy homomorphism on $A$ .

Proof.

Let $\{a_{i}\}_{i\in\{1,\dots,m\}}$ be a set of generators of $A$ . For each $i\in\{1,\dots,m\}$ , let $r_{i}$ denote $\vvvert a_{i}\vvvert_{\mathcal{B}}$ and let $\boldsymbol{r}$ denote the multi-radius consisting of $\{r_{i}\}_{i\in\{1,\dots,m\}}$ . Consider the Tate algebra $\mathcal{T}_{\boldsymbol{r}}$ and the homomorphism of $k$ -algebras

k[T_{1},\dots,T_{m}]\rightarrow\mathcal{B},\quad T_{i}\mapsto a_{i}

By the ultra-metricity of $\vvvert\mathord{\cdot}\vvvert_{\mathcal{B}}$ and the definition of $\boldsymbol{r}$ , one has

\forall n\in\mathbb{N},\forall J\in\mathbb{N}^{m},\forall f_{J}\in k,\vvvert\sum_{J\in\mathbb{N}^{m}}f_{J}\cdot\boldsymbol{a}^{J}\vvvert_{\mathcal{B}}\leq\vvvert\sum_{J\in\mathbb{N}^{m}}f_{J}\cdot\boldsymbol{a}^{J}\vvvert_{\mathcal{T}_{\boldsymbol{r}}}

so by a density argument one can extend it to a homomorphism of Banach $k$ -algebras

\mathcal{T}_{\boldsymbol{r}}\rightarrow\mathcal{B},\quad T_{i}\mapsto f_{i}

Let $\mathscr{I}$ be the kernel ideal of this homomorphism. To conclude it suffices to take $\mathcal{A}$ as $\mathcal{T}_{\boldsymbol{r}}/\mathscr{I}$ . ∎

Proposition 2.45.

Let $(B,\vvvert\mathord{\cdot}\vvvert)$ be a normed algebra and let $\mathcal{B}$ be its separated completion. Let $A$ be a sub- $k$ -algebra of $B$ , equipped with the restriction algebra norm of $\vvvert\mathord{\cdot}\vvvert$ , and let $\mathcal{A}$ be the separated completion of $(A,\vvvert\mathord{\cdot}\vvvert)$ . Assume that $\mathcal{A}$ is an affinoid algebra. If $B$ is integral and is finite over $A$ , then $\mathcal{B}$ is Banach finite over $\mathcal{A}$ . Therefore $\mathcal{B}$ is an affinoid algebra.

Proof.

By assumption, there exists $j\in\mathbb{N}$ and a homomorphism of $k$ -algebras and elements $\{e_{i}\}_{i\in\{1,\dots,j\}}\subseteq B$ such that

F:\bigoplus_{i\in\{1,\dots,j\}}A\rightarrow B,1_{i}\mapsto e_{i}

Moreover, $F$ is bounded

\vvvert\sum_{i\in\{1,\dots,j\}}a_{i}\cdot e_{i}\vvvert\leq\max_{i\in\{1,\dots,j\}}\vvvert a_{i}\cdot e_{i}\vvvert\leq\max_{i\in\{1,\dots,j\}}\vvvert e_{i}\vvvert\cdot\max_{i\in\{1,\dots,j\}}\vvvert a_{i}\vvvert

So $F$ extends to a homomorphism of Banach $\mathcal{A}$ -modules

\mathcal{F}:\bigoplus_{i\in\{1,\dots,j\}}\mathcal{A}\rightarrow\mathcal{B},1_{i}\mapsto e_{i}

Let $\mathcal{B}^{-}$ be the image of $\mathcal{F}$ , it is a Banach finite $\mathcal{A}$ -module with the quotient norm $\lVert\mathord{\cdot}\rVert_{\mathcal{F}}$ induced by $\mathcal{F}$ . As $\mathcal{B}^{-}$ is Banach finite over $\mathcal{A}$ , it is an affinoid algebra with an affinoid algebra spectral norm $\vvvert\mathord{\cdot}\vvvert^{-}$ , which is equivalent to $\lVert\mathord{\cdot}\rVert_{\mathcal{F}}$ . Now on $\mathcal{B}^{-}$ , $\vvvert\mathord{\cdot}\vvvert$ is bounded with respect to $\vvvert\mathord{\cdot}\vvvert^{-}$ by the continuity of $\mathcal{F}$ . To show the reverse, note that $\mathcal{B}^{-}$ is dense in $\mathcal{B}$ , so by Theorem 2.30 one has for any $b\in\mathcal{B}^{-}$

\vvvert b\vvvert^{-}=\max_{z\in\mathfrak{M}(\mathcal{B}^{-})}\lvert b(z)\rvert=\max_{z\in\mathfrak{M}(\mathcal{B})}\lvert b(z)\rvert=\vvvert b\vvvert_{\mathrm{sp}}\leq\vvvert b\vvvert

Therefore $\vvvert\mathord{\cdot}\vvvert$ and $\vvvert\mathord{\cdot}\vvvert^{-}$ are equivalent norms on $\mathcal{B}^{-}$ , so $\mathcal{B}^{-}$ is closed in $\mathcal{B}$ , hence coincides with it. ∎

To make an affinoid algebra strict, one can enlarge the base field.

Lemma 2.46.

Let $\boldsymbol{r}=(r_{1},\dots,r_{n})$ be a multi-radius such $\{\alpha(\log r_{i})\}_{i\in\{1,\dots,n\}}$ are $\mathbb{Q}$ -linearly independent. Then the $k$ -affinoid algebra

K_{\boldsymbol{r}}:=k\{\boldsymbol{r}^{-1}\boldsymbol{T},\boldsymbol{r}\boldsymbol{T}^{-1}\}=k\{\boldsymbol{r}^{-1}\boldsymbol{T},\boldsymbol{r}\boldsymbol{S}\}/(T_{1}S_{1}-1,\dots,T_{n}S_{n}-1)

is a field. ([Ber, Definition 2.1.1])

Lemma 2.47.

Let $\mathcal{T}_{n}(\boldsymbol{r})$ be a $k$ -Tate alegbra. It is strict if and only if $r_{i}\in\sqrt{\lvert k^{\times}\rvert}$ for all $i$ . ([Ber, Corollary 2.1.6])

Corollary 2.48.

Let $\mathcal{T}_{n}(\underline{r})$ be a $k$ -Tate alegbra. Let $I\subseteq\{1,\dots,n\}$ be a subset of indices such that $\{\alpha(\log r_{i})\}_{i\in I}$ are $\mathbb{Q}$ -linearly independent and $|I|$ is maximal for this independence property. Let $\boldsymbol{r}_{I}=(r_{i_{1}},\dots,r_{i_{1}})$ , then $K_{\boldsymbol{r}_{I}}\widehat{\otimes}_{k}\mathcal{T}_{n}(\boldsymbol{r})$ is a strict $K_{\boldsymbol{r}_{I}}$ -Tate algebra.

Corollary 2.49.

For any $k$ -affinoid algebra $\mathcal{A}$ , there exists a multi-radius $\boldsymbol{r}_{I}=(r_{i})_{i\in I}$ such that $\{\alpha(\log r_{i})\}_{i\in I}$ are $\mathbb{Q}$ -linearly independent and $K_{\boldsymbol{r}_{I}}\widehat{\otimes}_{k}\mathcal{A}$ is a $K_{\boldsymbol{r}_{I}}$ -strict affinoid algebra. ([Ber, Proposition 2.1.2])

2.3.2. Algebraic structures: Noetherianity

Let $\mathcal{A}$ be a Banach $k$ -algebra, one denotes by $\mathcal{A}^{\circ}$ the $k^{\circ}$ -algebra $\{f\in\mathcal{A}\text{ }|\text{ }\vvvert f\vvvert_{\mathcal{A},\text{sp}}\leq 1\}$ , and by $\mathcal{A}^{\circ\circ}$ the ideal of $\mathcal{A}^{\circ}$ constituting of elements $\vvvert f\vvvert_{\mathcal{A},\text{sp}}<1$ . The $\widetilde{k}$ -algebra $\mathcal{A}^{\circ}/\mathcal{A}^{\circ\circ}$ is called the reduction of $\mathcal{A}$ . It can be shown that $\widetilde{\mathcal{T}_{n}}$ is isomorphic to $\widetilde{k}[T_{1},\dots,T_{n}]$ . ([BGR, Proposition 5.1.2.2])

Definition 2.50.

An element $f\in\mathcal{T}_{n}$ with $\vvvert f\vvvert_{\mathcal{T}_{n}}=1$ is said to be regular in $z_{n}$ of degree $d$ if its reduction $\bar{f}=\lambda(z_{n})^{d}+\sum_{0\leq i\leq d-1}c_{i}(z_{n})^{d-i}$ in $\bar{\mathcal{T}_{n}}$ where $\lambda\in k^{\times}$ and $c_{i}\in\bar{k}[z_{1},\dots,z_{n-1}]$ .

Proposition 2.51.

[Weierstrass division] Let $\mathcal{T}_{n}$ be the $k$ -Tate algebra of multiradius $\underline{r}=\underline{1}$ , then

(1)

Let $f\in\mathcal{T}_{n}$ be an distinguished element in $z_{n}$ of degree $d$ , and $g\in\mathcal{T}_{n}$ be any element. Then there exist unique $r\in\mathcal{T}_{n-1}[z_{n}]$ of degree less than $d$ in $z_{n}$ and $q\in\mathcal{T}_{n}$ such that $g=q\cdot f+r$ . Moreover $\vvvert g\vvvert_{\mathcal{T}_{n}}=\max\{\vvvert q\vvvert_{\mathcal{T}_{n}},\vvvert r\vvvert_{\mathcal{T}_{n}}\}$
(2)

Let $f\in\mathcal{T}_{n}$ with $\vvvert f\vvvert_{\mathcal{T}_{n}}=1$ . Then there exists a $k$ -algebra automorphism $\tau$ of $\mathcal{T}_{n}$ such that $\tau(f)$ is regular in $z_{n}$ .

([BGR, Theorem 5.2.1.2], [FvdP, Theorem 3.1.1])

Proposition 2.52.

The Tate algebra $\mathcal{T}_{n}$ is Noetherian. All of its ideals are closed. ([BGR, Theorem 5.2.6.1, Corollary 5.2.7.2], [FvdP, Theorem 3.2.1])

Corollary 2.53.

Any strict affinoid algebra is Noetherian. All of its ideals are closed ([BGR, Proposition 6.1.1.3], [FvdP, Theorem 3.2.1]). Any affinoid algebra is Noetherian. All of its ideals are closed ([Ber, Propositon 2.1.3]).

Proposition 2.54.

[Noether normalization] For strict affinoid algebra $\mathcal{A}$ , there exists an injective finite and admissible Banach algebra homomorphism $\mathcal{T}_{d}\to\mathcal{A}$ for some $d>0$ . Moreover, $d$ equals the Krull dimension of $\mathcal{A}$ . ([BGR, Theorem 6.1.2.1], [FvdP, Theorem 3.2.1])

Corollary 2.55.

Let $\mathfrak{m}$ be a maximal ideal of strict affinoid algebra $\mathcal{A}$ , then $\mathcal{A}/\mathfrak{m}$ is a finite extension of $k$ .

2.3.3. Topological structures: the spectral norm

The Gauss norm on Tate algebra is equal to its spectral norm. For a general strict redueced affinoid algebra, the Banach algebra norm is equivalent to its spectral seminorm, thanks to the compatibility of Banach algebra norms with algebraic structures.

One studies the spectral norm of the Tate algebra case by direct calculation.

Proposition 2.56.

For any $f\in\mathcal{T}_{n}$ , there exists $z\in\mathrm{Max}(\mathcal{T}_{n})$ such that $\lvert f(z)\rvert_{z}=\vvvert f\vvvert_{\mathcal{T}_{n}}$ ([BGR, Proposition 5.1.4.3]). On $\mathcal{T}_{n}$ , the three norms are equal: $\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n}}=\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n},\text{sp}}=\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n},\mathrm{spM}}$ .

One then uses Noether normalization to investigate the spectral seminorm of general affinoid algebra.

Proposition 2.57.

Let $\mathcal{A}$ be a reduced strict affinoid algebra. Then its spectral norm $\vvvert\mathord{\cdot}\vvvert_{\mathcal{A},\text{sp}}$ is a complete norm on $\mathcal{A}$ . It is equivalent to the Banach algebra norm $\vvvert\mathord{\cdot}\vvvert_{\mathcal{A}}$ . ([FvdP, Theorem 3.4.9], [BGR, Theorem 6.2.4.1])

Corollary 2.58.

Let $\mathcal{A}$ be a reduced general affinoid algebra. Then there exists $C>0$ such that $\vvvert f\vvvert\leq C\vvvert f\vvvert_{\mathrm{sp}}$ for all $f\in\mathcal{A}$ . In particular, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is complete on $\mathcal{A}$ , and is equivalent to $\vvvert\mathord{\cdot}\vvvert$ . ([Ber, Proposition 2.1.4.ii])

Remark 2.59.

The constant $C$ here does not depend on $f\in\mathcal{A}$ , it is uniform.

2.3.4. Affinoid space as locally ringed space

The Berkovich spectrum of affinoid algebras are called affinoid spaces. It is possible to put locally ringed space structures on them. The construction of structural sheaf goes first with a Grothendieck topology generated by closed compact subsets of affinoid domains, then passes to the canonical topology by a limit process approximating an open set by these compact sets.

Affinoid domains and structural algebra

Definition 2.60.

Let $\mathcal{A}$ be an affinoid algebra. An affinoid domain is a closed subset $V$ of $\mathfrak{M}(\mathcal{A})$ , which is homeomorphic to $(\iota_{V})^{\star}(\mathfrak{M}(\mathcal{A}_{V}))$ for some affinoid algebra $\mathcal{A}_{V}$ and Banach algebra homomorphism $\iota_{V}:\mathcal{A}\to\mathcal{A}_{V}$ , and satisfies the universal mapping property: for any Banach algebra homomorphism $\phi:\mathcal{A}\to\mathcal{C}$ between affinoid algebras with $\phi^{\star}(\mathfrak{M}(\mathcal{C}))\subseteq V$ , there exists a unique Banach algebra homomorphism $\psi:\mathcal{A}_{V}\to\mathcal{C}$ with $\phi=\psi\circ\iota_{V}$

Lemma 2.61.

Let $V$ be an affinoid domain in $\mathfrak{M}(\mathcal{A})$ . Then $V$ is homeomorphic to $\mathfrak{M}(\mathcal{A}_{V})$ . Moreover $\mathcal{A}_{V}$ is a flat $\mathcal{A}$ -algebra. ([Ber, Proposition 2.2.4])

Example 2.62.

Given $f=(f_{1},\dots,f_{m})$ and $g=(g_{1},\dots,g_{n})$ tuples of elements of $\mathcal{A}$ , $p=(p_{1},\dots,p_{m})\in(\mathbb{R}_{+}^{*})^{m}$ and $q=(q_{1},\dots,q_{n})\in(\mathbb{R}_{+}^{*})^{n}$ , the closed subset

V=\mathfrak{M}(\mathcal{A})(p^{-1}f,qg^{-1}):=\{z\in\mathfrak{M}(\mathcal{A}),|f_{i}(z)|_{z}\leq p_{i},\text{ }|g_{j}(z)|_{z}\geq q_{j}\}

is an affinoid domain. The corresponding homomorphism of affinoid algebras is

\mathcal{A}\to\mathcal{A}_{V}=\mathcal{A}\{p_{1}^{-1}T_{1},\dots,p_{m}^{-1}T_{m},q_{1}S_{1},\dots,q_{n}S_{n}\}/(T_{i}-f_{i},g_{j}S_{j}-1)

Such domains are called Laurent domains. If $n=0$ , they are called Weierstrass domains.

Lemma 2.63.

A finite intersection of affinoid domains is an affinoid domain. ([Ber, Remark 2.2.2.iv])

Corollary 2.64.

Any point $z\in\mathfrak{M}(\mathcal{A})$ has a fundamental system of (closed) neighbourhoods consisting of affinoid domains. ([Ber, Proposition 2.2.3])

Special domains and acyclicity of structural presheaf

Definition 2.65.

A special domain $V$ in $\mathfrak{M}(\mathcal{A})$ is a finite union of affinoid domains $V_{i}$ in $\mathfrak{M}(\mathcal{A})$ .

Definition 2.66.

The Grothendieck topology on $\mathfrak{M}(\mathcal{A})$ is the one with special domains as admissible open sets and finite covering as admissible coverings. One notes $\mathfrak{M}(\mathcal{A})_{G}$ for the space with this G-topology.

Definition 2.67.

Let $\mathfrak{V}$ be an admissible covering of $\mathfrak{M}(\mathcal{A})$ by affinoid domains $\{V_{i}\}_{i\in I}$ , where $I$ is a finite set. Then for a Banach finite $\mathcal{A}$ -module $\mathcal{M}$ , the Cech complex of $\mathcal{M}$ with respect to $V_{i}$ is defined to be the complex of Banach $\mathcal{A}$ -modules

C^{\centerdot}(\mathcal{M},\mathfrak{V}):\text{ }0\to\mathcal{M}\to\prod_{i\in I}\mathcal{M}_{i}\to\prod_{i,j\in I}\mathcal{M}_{i,j}\to\dots

One would like to have acyclicity of the complex $C^{\centerdot}(\mathcal{M},\mathfrak{V})$ in order to follow standard construction of a structural sheaf on $\mathfrak{M}(\mathcal{A})_{G}$ .

Theorem 2.68.

Let $\mathcal{A}$ be a strict affinoid algebra and $\mathfrak{V}$ an admissible covering by strict affinoid domains for $\mathfrak{M}(\mathcal{A})$ . Then $C^{\centerdot}(\mathcal{A},\mathfrak{V})$ is acyclic. ([BGR, Proposition 8.2.2.5])

Corollary 2.69.

For general affinoid domain $\mathfrak{M}(\mathcal{A})$ with general affinoid domains covering $\mathfrak{V}$ , the complex $C^{\centerdot}(\mathcal{A},\mathfrak{V})$ is acyclic. So is $C^{\centerdot}(M,\mathfrak{V})$ for finite Banach $\mathcal{A}$ -module $M$ . ([Ber, Proposition 2.2.5])

Definition 2.70.

Let $V$ be any special domain in $\mathfrak{M}(\mathcal{A})$ . Fix a way of writing $V$ as $\bigcup_{i\in I}V_{i}$ where $I$ is a finite set and $V_{i}=\mathfrak{M}(\mathcal{A}_{V_{i}})$ are affinoid algebras, let

\mathcal{A}_{V}:=\ker(\prod_{i\in I}\mathcal{A}_{V_{i}}\to\prod_{i,j\in I}\mathcal{A}_{V_{i}\cap V_{j}})

be the $k$ -Banach algebra with sub-norm. The structural pre-sheaf of affinoid algebras $\mathscr{O}_{\mathfrak{M}(\mathcal{A})_{G}}$ on $\mathfrak{M}(\mathcal{A})_{G}$ (with respect to the G-topology) is the one assigning $V$ the $k$ -Banach algebra $\mathcal{A}_{V}$ . It is a sheaf thanks to Corollary 2.69.

Remark 2.71.

The $k$ -Banach algebra $\mathscr{O}_{\mathfrak{M}(\mathcal{A})_{G}}(V)$ does not depend on the way of being a union of affinoid domains.

Definition 2.72.

For any open subset $U$ of $\mathfrak{M}(\mathcal{A})$ , let $\mathscr{O}_{\mathfrak{M}(\mathcal{A})}$ be the pre-sheaf of $k$ -algebras (with respect to the canonical topology) which assigns $U$ the limit

\mathscr{O}_{\mathfrak{M}(\mathcal{A})}(U):=\varprojlim_{V\subset U,\text{V special domain}}\mathcal{A}_{V}

It is also a sheaf thanks to the compactness of special domains under canonical topology. This is called the structural sheaf of $\mathfrak{M}(\mathcal{A})$ .

Proposition 2.73.

$\mathscr{O}_{\mathfrak{M}(\mathcal{A})}$ is a sheaf of local rings. The topological space $\mathfrak{M}(\mathcal{A})$ has a structure of locally ringed space given by the sheaf $\mathscr{O}_{\mathfrak{M}(\mathcal{A})}$ . ([Ber, Section 2.3])

2.4. Spectral calculus

Gelfand-Shilov theory allows one to do multi-variable spectral calculus for (commutative) Banach algebras over $\mathbb{C}$ . In particular, one can localize a homomorphism between Banach algebras onto a neighbourhood of its spectrum. Similar theory, as develloped in [Ber, Chapter 7], exists in the non-Archimedean base field setting.

2.4.1. Holomorphic envelop

The holomorphic convexity of spectrum of a homomorphism of Banach $k$ -algebra depends on the dense-ness of its image. In case where the spectrum of a homomorphism is not holomorphic convex, one can add variables to the source algebra so that spectrum of extended homomorphism is holomorphically convex.

Definition 2.74.

Let $\mathcal{A}$ and $\mathcal{B}$ be Banach $k$ -algebras, and $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach algebras. The spectrum of homomorphism $\phi$ is the image of $\mathfrak{M}(\mathcal{B})$ in $\mathfrak{M}(\mathcal{A})$ under $\phi^{\star}$ . Denote it by $\Sigma_{\phi}$

Definition 2.75.

Let $\mathcal{A}$ be a Banach $k$ -algebra. Let $\Omega$ be a compact subset of $\mathfrak{M}(\mathcal{A})$ . The holomorphic convex envelop of $\Omega$ in $\mathfrak{M}(\mathcal{A})$ is the subset

\Omega^{\mathrm{h}}:=\{z\in\mathfrak{M}(\mathcal{A})\ |\ \forall f\in\mathcal{A},\lvert f\rvert_{z}\leq\sup_{z^{\prime}\in\Omega}\lvert f\rvert_{z^{\prime}}\}

The subset $\Omega$ is said to be holomorphically convex if $\Omega^{\mathrm{h}}=\Omega$ .

Lemma 2.76.

The intersection of all Weierstrass neighbourhoods of $\Omega$ in $\mathfrak{M}(\mathcal{A})$ coincide with $\Omega^{\mathrm{h}}$ . ([Ber, Proposition 2.6.1])

Proposition 2.77.

Let $\mathcal{A}$ be a $k$ -affinoid algebra, $\mathcal{B}$ be a Banach $k$ -algebra. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach $k$ -algebras. Let $\mathcal{B}^{\prime}$ be the closed sub-algebra generated by the image of $\phi$ of $\mathcal{A}$ in $\mathcal{B}$ and let $\phi^{\prime}:\mathcal{A}\to\mathcal{B}^{\prime}$ be the restricted homomorphism. Then $(\Sigma_{\phi})^{\mathrm{h}}=\Sigma_{\phi^{\prime}}$ . ([Ber, Proposition 7.3.1])

Corollary 2.78.

Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach algebras from an affinoid algebra to a Banach algebra with dense image. Then $\Sigma_{\phi}$ is holomorphically convex.

One has the following analogue of Arens-Calderon theorem, which holomorphically convexifies the spectrum of a homomorphism of Banach $k$ -algebras by adding variables on the source algebra.

Proposition 2.79.

Let $\mathcal{A}$ be a $k$ -affinoid algebra, $\mathcal{B}$ be a Banach $k$ -algebra. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach $k$ -algebras. Then for any open neighbourhood $U$ in $\mathfrak{M}(\mathcal{A})$ of the spectrum $\Sigma_{\phi}$ , there exists a homomorphism of Banach algebras extending $\phi$

\widetilde{\phi}:\widetilde{\mathcal{A}}:=\mathcal{A}\{r_{1}^{-1}T_{1},\dots,r_{n}^{-1}T_{n}\}\to\mathcal{B}

such that $\mathrm{pr}((\Sigma_{\phi})^{\mathrm{h}})\subseteq U$ , where $\mathrm{pr}:\mathfrak{M}(\widetilde{\mathcal{A}})\to\mathfrak{M}(\mathcal{A})$ is the canonical map of projection. ([Ber, Proposition 7.3.3])

2.4.2. Holomorphic functional calculus

It is easy to localize the homomorphism to holomorphic convex neighbourhood of its spectrum. For a spectrum of homomorphism which is not holomorphically convex, one uses Proposition 2.79 to localize the homomorphism to any neighbourhood of it.

Lemma 2.80.

Let $\phi:\mathcal{A}\to\mathcal{B}$ be a Banach algebra homomorphism from an affinoid algebra $\mathcal{A}$ to a Banach algebra $\mathcal{B}$ . Then for any Laurent domain neighbourhood $V$ of $\Sigma_{\phi}$ , $\phi$ extends to a unique Banach algebra homomorphism $\phi_{V}:\mathcal{A}_{V}\to\mathcal{B}$ . ([Ber, Corollary 2.5.16])

Theorem 2.81.

Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach algebras from an affinoid algebra to a Banach algebra. Let $V\subseteq\mathfrak{M}(\mathcal{A})$ be any special domain containing $\Sigma_{\phi}$ . Then there exists a Banach algebra homomorphism

\theta_{\phi}:\Gamma(V,\mathscr{O}_{\mathfrak{M}(\mathcal{A})})\to\mathcal{B}

satisfying $\phi=\theta_{\phi}\circ\iota_{V}$ , where $\iota_{V}:\mathcal{A}\to\mathcal{A}_{V}=\Gamma(V,\mathscr{O}_{\mathfrak{M}(\mathcal{A})})$ is the Banach algebra homomorphism corresponding to the inclusion $V\subseteq\mathfrak{M}(\mathcal{A})$ . ([Ber, Theorem 7.3.4])

Remark 2.82.

One can verify that the resulting Banach algebra homomorphism does not depend on the choice of $\widetilde{\phi}$ .

2.5. Analytification of scheme of finite type

There is a construction of Berkovich spectrum for a $k$ -algebra similar to the one for $k$ -Banach algebra, giving rise to analytification of $k$ -schemes of locally finite type, as developped in [Ber, Section 3.4].

2.5.1. Local situation

For affine varieties, the topological space of its analytification is defined in the same way as the spectrum of Banach algebra, except that boundedness requirement of seminorms are dropped. They enjoy similar basic properties as the spectrum of Banach algebra. Proofs are of same spirit hence are omitted.

Definition 2.83.

Let $Z=\spec(A_{Z})$ be an affine $k$ -scheme of finite type, where $A_{Z}$ is a $k$ -algebra of finite type. Its Berkovich analytification $Z^{an}$ is the topological space constituting of all multiplicative seminorms $\vvvert\mathord{\cdot}\vvvert$ on $A_{Z}$ as points and with the canonical topology (the weakest topology making every function $\vvvert\mathord{\cdot}\vvvert\to\vvvert f\vvvert$ continuous for each $f\in A_{Z}$ ). ([Ber, Remark 3.4.2])

Definition 2.84.

A character on $A_{Z}$ is a homomorphism of $k$ -algebra from $A_{Z}$ to some valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ over $(k,\lvert\mathord{\cdot}\rvert_{k})$ . Two characters $\chi_{1}:A_{Z}\to K_{1}$ and $\chi_{1}:A_{Z}\to K_{1}$ are called equivalent if there exists a $k$ -algebra homomorphism $\chi_{3}:A_{Z}\to K_{3}$ and norm preserving $k$ -algebra homomorphisms $i_{1}:K_{1}\to K_{3}$ and $i_{2}:K_{2}\to K_{3}$ satisfying $\chi_{3}=i_{1}\circ\chi_{1}=i_{2}\circ\chi_{2}$ .

Lemma 2.85.

There is a bijective map from the set of points of $Z^{an}$ to the set of equivalent classes of characters on $A_{Z}$ .

Proposition 2.86.

Let $\phi:A_{Z}\to A_{W}$ be a homomorphism of $k$ -algebras of finite type where $Z=\spec(A_{Z})$ and $W=\spec(A_{W})$ . Then there is an induced continuous map $\phi^{\star}:W^{an}\to Z^{an}$ , which sends a multiplicative seminorm $|\cdot|_{w}$ to $|\phi(\cdot)|_{w}$ .

Proposition 2.87.

If $\phi$ is surjective, then $\phi^{\star}$ is injective and is a closed map; if $\phi$ is finite, then $\phi^{\star}$ is surjective. ( [Ber, Proposition 3.46 (6)(7)])

Proposition 2.88.

Let $Z=\spec(A_{Z})$ be an affine $k$ -variety, $\vvvert\mathord{\cdot}\vvvert$ an algebra norm on $A_{Z}$ and $\mathcal{A}_{Z}$ be the $k$ -Banach algebra obtained by completing $A_{Z}$ with respect to $\vvvert\mathord{\cdot}\vvvert$ . Then the canonical homomorphism of $k$ -algebras from $A_{Z}$ to $\mathcal{A}_{Z}$ induces a continuous map which embeds the Berkovich spectrum $\mathfrak{M}(\mathcal{A}_{Z})$ into $Z^{an}$ as a compact subspace (and is closed since $Z^{\mathrm{an}}$ is Hausdorff), and the Berkovich topology coincides with the induced topology from $Z^{an}$ .

Proof.

For any $z\in\mathfrak{M}(\mathcal{A}_{Z})$ , the multiplicative algebra seminorm (or the corresponding character) $\chi_{z}$ on $\mathcal{A}_{Z}$ corresponds to a unique multiplicative algebra seminorm on $A_{Z}$ by restriction. Since $A_{Z}$ is dense in $\mathcal{A}_{Z}$ , the family of open sets $\{U(f;p,q),\text{ }f\in A_{Z},\text{ }p,q\in\mathbb{R}\}$ form a basis for topology on $\mathfrak{M}(\mathcal{A}_{Z})$ , hence the inherited topology coincides with the originial topology. So the embedding is continuous, and the image of $\mathfrak{M}(\mathcal{A}_{Z})$ is compact in $Z^{an}$ . Since the topology on $Z^{an}$ is Hausdorff, the image of $\mathfrak{M}(\mathcal{A}_{Z})$ is closed. ∎

Definition 2.89.

An analytic function on open set $U\subseteq(\spec A_{Z})^{an}$ is a map $h:U\to\coprod_{z\in U}\hat{\kappa}(z)$ which is a local uniform limit of rational functions: every $z\in U$ has an open neighbourhood $U^{\prime}\subseteq U$ such that for every $\epsilon>0$ , there exists $f_{U^{\prime}},g_{U^{\prime}}\in A_{Z}$ with $|h(z)-\frac{f_{U^{\prime}}(z)}{g_{U^{\prime}}(z)}|<\epsilon$ and $g(z)\neq 0$ for all $z\in U^{\prime}$ . Denote by $\mathcal{R}^{\mathrm{an}}(U)$ the $k$ -algebra of all analytic functions on $U$ .

Definition 2.90.

The structural sheaf $\mathscr{O}_{Z^{\mathrm{an}}}$ on $Z^{\mathrm{an}}$ is the one assigning $\mathcal{R}^{\mathrm{an}}(U)$ to an open set $U$ .

Proposition 2.91.

$\mathscr{O}_{Z^{\mathrm{an}}}$ is a sheaf of local rings. The pair $(Z^{\mathrm{an}},\mathscr{O}_{Z^{\mathrm{an}}})$ gives rise to a locally ringed space.

Proposition 2.92.

If $\mathcal{A}_{Z}$ is an affinoid algebra $\mathcal{A}_{Z}$ , then there is a morphism of locally ringed space

(\mathfrak{M}(\mathcal{A}_{Z},\mathscr{O}_{\mathfrak{M}(\mathcal{A}_{Z})})\to(Z^{an},\mathscr{O}_{Z^{an}})

Proof.

The map of topological spaces is given in Proposition 2.88. For the ring homomorphism, it suffices to construct a $k$ -algebra homomorphism $\mathscr{O}_{Z^{an}}(U)\to\mathcal{A}_{V}$ for any open set $U\subseteq\mathfrak{M}(\mathcal{A})$ and any affinoid domain $V\subseteq U$ . Moreover, it suffices to consider $U$ and $V$ of basic form

U=U(\underline{p}^{-1}\underline{f},\underline{q}\underline{g}^{-1}),\quad V=\mathfrak{M}(\mathcal{A}_{Z}((\underline{p}-\underline{\epsilon})^{-1}\underline{f},(\underline{q}+\underline{\epsilon})\underline{g}^{-1}))\text{ , }\epsilon>0

There is a homomorphism of $k$ -algebras $\mathcal{R}^{\mathrm{an}}(U)\to\mathcal{A}_{Z}((\underline{p}-\underline{\epsilon})^{-1}\underline{f},(\underline{q}+\underline{\epsilon})\underline{g}^{-1})$ sending $\frac{\tilde{f}}{\tilde{g}}$ for $\tilde{f},\tilde{g}\in A_{Z}$ to itself, the later being an element of $\mathcal{A}_{V}$ since $\frac{1}{\tilde{g}}\in\mathcal{A}_{V}$ by Lemma 2.25. As uniform limits of sequence in $\mathcal{R}^{\mathrm{an}}(U)$ remains to be uniform limits, this homomorphism extends to a $k$ -algebra homomorphism $\mathscr{O}_{Z^{an}}(U)\to\mathcal{A}_{V}$ . ∎

2.5.2. Global situation

One can analytify a scheme of finite type defined over $k$ by glueing local constructions.

Definition 2.93.

Let $X$ be a finite type scheme over $\spec k$ , and write $X$ as $\bigcup X_{i}$ where $X_{i}=\spec A_{X_{i}}$ are affine charts. The Berkovich analytification of $(X,\mathscr{O}_{X})$ is the locally ringed space obtained by gluing the Berkovich analytification $((X_{i})^{an},\mathscr{O}_{(X_{i})^{an}})$ of each $(X_{i},\mathscr{O}_{X_{i}})$ .

Proposition 2.94.

Let $\phi:X\rightarrow Y$ be a morphism of schemes of locally finite type over $\spec k$ . Then it induces a continuous map $\phi^{\mathrm{an}}:X^{\mathrm{an}}\rightarrow Y^{\mathrm{an}}$ . And $\phi$ is (1) separated, (2) injective, (3) surjective, (4) an open immersion and (5) an isomorphism if and only if $\phi^{\mathrm{an}}$ has the same property. ([Ber, Proposition 3.4.6])

Theorem 2.95.

If $X$ is proper, then $X^{\mathrm{an}}$ is Hausdorff and compact. ([Ber, Theorem 3.4.8])