Affinoid domains and structural algebra [02GE]

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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])