2.2.2. Spectrum [00MM]

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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])