2.3. Affinoid algebras [00MR]

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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])