2.3.4. Affinoid space as locally ringed space [00MV]

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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])