3 Analytification of a scheme

3.1 Toplogical story

3.1.1 Affine scheme

Let $S$ be a finitely generated $\mathbb {C}$-algebra so that $S \cong \mathbb {C}[a_{1},\dots , a_{n}]$ for some $a_{i} \in S$. Thus there is a surjection $\theta : \mathbb {C}[X_{1}, \dots , X_{n}] \to S$ defined by $X_{i} \mapsto a_{i}$. Thus, we have a morphism $\left(\operatorname *{Spec}{\theta }, \widetilde{\theta }\right)$ of schemes of finite type over $\mathbb {C}$ between $\left(\operatorname *{Spec}{S}, \widetilde{S}\right)$ to $\left(\operatorname *{Spec}{\mathbb {C}[X_{1},\dots , X_{n}]}, \widetilde{\mathbb {C}[X_{1},\dots , X_{n}]}\right)$. By Proposition 2.24, we know that $\operatorname {Spec}\theta $ gives us a continuous map

\[ \operatorname {MaxSpec}\theta : \operatorname {MaxSpec} S \to \operatorname {MaxSpec} \mathbb {C}[X_{1},\dots , X_{n}], \]

since $\theta $ is surjective, $\operatorname {MaxSpec}\theta $ is injective ¹ .

Theorem 3.1

The set of closed points in $\operatorname {Spec}{\mathbb {C}[X_{1},\dots , X_{n}]}$ corresponds bijectively to $\mathbb {C}^{n}$.

Proof ▶

By Proposition 2.21, the set of closed points bijects to $\mathbb {C}$-algebra homomorphisms $\mathbb {C}[X_{1}, \dots , X_{n}] \to \mathbb {C}$. Thus we only need a bijection between $\mathbb {C}$-algebra homomorphism $\mathbb {C}[X_{1}, \dots , X_{n}] \to \mathbb {C}$ and $\mathbb {C}^{n}$:

Give a point $p := (a_{1},\dots , a_{n}) \in \mathbb {C}^{n}$, we define $\phi _{a}: \mathbb {C}[X_{1},\dots , X_{n}] \to \mathbb {C}$ to be evaluation at the point $p$.
Give a $\mathbb {C}$-algebra homomorphism $\phi $, we take the point to be $(\phi (X_{1}, \dots , X_{n}))$.

Definition 3.2 Analytification of topological spaces.

The complex topology of $\operatorname {Spec}S$ is the subspace topology of $\mathbb {C}^{n}$ via the injective map $\operatorname {MaxSpec}\theta $. With the complex topology, we denote $\operatorname {MaxSpec}S$ as ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}$ and, if $\phi : S \to S'$ is a $\mathbb {C}$-algebra homorphism, we the induced map between ${\left\{ {\operatorname {Spec}{S’}}\right\} }^{\mathsf{an}}$ and ${\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$ as ${\left\{ {\operatorname {Spec}\phi }\right\} }^{\mathsf{an}}$.

Note that by now we do not know that ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}$ is independent from the choice of generators $\{ a_{1},\dots , a_{n}\} $, we will enventually prove that this is true, but let’s write ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}_{a_{i}}$ to stress the dependency.

Theorem 3.3

If $S$, as $\mathbb {C}$-algebras, is generated by both $a_{1},\dots , a_{n}$ and $b_{1},\dots ,b_{m}$, we would have as topological spaces ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}_{a_{i}}$ and ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}_{b_{i}}$ are homeomophic.

Proof ▶

Let us abbreviate the polynomial rings $\mathbb {C}[X_{1},\dots , X_{n}]$ as $R$ and $\mathbb {C}[Y_{1},\dots , Y_{m}]$ as $R'$, then we have two surjective homomorphisms $\theta : R \to S$ and $\theta ' : R' \to S$ such that $\theta (X_{i}) = a_{i}$ and $\theta '(Y_{i})=b_{i}$.

It is sufficient, by symmetry, to prove topology induced by generators $b_{i}$’s is finer than that of $a_{i}$’s.

Since $b_{i}$’s generate $S$ and $a_{i}$’s are in $S$, we can find $n$ polynomials $P_{i} \in R' = \mathbb {C}[Y_{1},\dots , Y_{m}]$ such that $a_{i} = P_{i}(b_{1},\dots , b_{m})$. Thus we can define $\phi : R \to R'$ by $X_{i}\mapsto P_{i}(Y_{1},\dots , Y_{m})$ such that $\theta = \theta ' \circ \phi $. Thus we have a commutative diagram (of plain functions)

$\begin{tikzcd} [column sep=large] {\an{\left(\specop{S}\right)}_{a_{i}}} \arrow{r}{\maxspecop\theta} \arrow[leftrightarrow]{d}[left]{=} & \complex^{n} \\ {\an{\left(\specop{S}\right)}_{b_{i}}} \arrow{r}{\maxspecop{\theta'}} & \complex^{m} \arrow{u}[right]{\maxspecop{\phi}} \end{tikzcd}$

It is sufficient to prove that $\operatorname {MaxSpec}\phi $ is continuous, then since $\operatorname {MaxSpec}\theta $, $\operatorname {MaxSpec}\theta '$ and $\operatorname {MaxSpec}\phi $ are all continuous, the identity function ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}_{a_{i}} \to {\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}_{b_{i}}$ is continuous. Consider a point $c = (c_{1},\dots , c_{m}) \in \mathbb {C}^{m}$, then $\operatorname {MaxSpec}\phi (c)$ is the point $\mathrm{eval}_{c} \circ \phi (X_{1},\dots , X_{n})$, i.e. $(c_{1},\dots , c_{m}) \mapsto (P_{1}(c_{1},\dots ,c_{m}),\dots , P_{n}(c_{1},\dots ,c_{m}))$. This is a map defined by polynomials, thus is continuous.

Now we have proven that the complex topology is independent of generators, we can write ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}$ with a clear conscience.

Lemma 3.4

Since ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}$, as a set, is just the set of closed points of $\operatorname {Spec}S$, we have a function $\lambda : {\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}} \hookrightarrow \operatorname {Spec}S$. $\lambda $ is continuous where ${\left\{ {\left(\operatorname {Spec}S\right)}\right\} }^{\mathsf{an}}$ is with complex topology while $\operatorname {Spec}S$ is with the Zaraski toplogy.

Proof ▶

Let us choose a set of generators $a_{1},\dots , a_{n}$ and write $R := \mathbb {C}[X_{1},\dots , X_{n}]$, then we would have the following commutative diagram:

$\begin{tikzcd} \an{\left(\specop{S}\right)} \arrow[hookrightarrow]{r}{\lambda_{S}} \arrow[red]{d}[left]{\mathrm{restriction~of}~\specop{\theta}}& \specop{S} \arrow{d}{\specop{\theta}} \\ \an{\left(\specop{R}\right)} \arrow[hookrightarrow]{r}{\lambda_{R}} \arrow[red]{d}[left]{\sim} & \specop{R} \\ \complex^{n} & \end{tikzcd}$

where $\theta $ is the surjective $\mathbb {C}$-algebra homomorphism $R \to S$. The red arrows are continuous, since they define the complex topology; $\operatorname {Spec}\theta $ is continuous as well. To prove $\lambda _{S}$ is continuous, we only need to prove the special case $\lambda _{R}$ where $R = \mathbb {C}[X_{1},\dots , X_{n}]$. Since $\operatorname {Spec}R$ has a basis of basic open set $D(f)$, we only need to check that $D(f) \cap {\left\{ {\left(\operatorname {Spec}R\right)}\right\} }^{\mathsf{an}}$ is open for any polynomial $f \in \mathbb {C}[X_{1}, \dots , X_{n}]$, indeed the intersection is equal to $\{ x \in \mathbb {C}^{n}| f(x) \ne 0 \} $ thus open ² ³ .

Lemma 3.5

Let $a_{1},\dots , a_{n}$ be a set of generators of $S$ as $\mathbb {C}$-algebra and $R$ be $\mathbb {C}[X_{1},\dots ,X_{n}]$ and $\theta : R \to S$ be the surjective map defined by $\theta (X_{i})=a_{i}$. The image of $\operatorname {Spec}{\theta } : {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}} \to {\left\{ {\operatorname {Spec}{R}}\right\} }^{\mathsf{an}}\cong \mathbb {C}^{n}$ is

\[ V(\ker \theta ):=\{ (x_{1},\dots ,x_{m})| f_{i}(x_{1},\dots ,x_{n})=0\} = \{ (x_{1},\dots ,x_{m})| p(x_{1},\dots ,x_{n})=0~ \text{for all}~ p\in \ker {\theta }\} , \]

where $f_{i}$ generates $\ker \theta $.

Proof ▶

$x = (x_{1},\dots ,x_{n})\in \operatorname {image}{\left\{ {\operatorname {Spec}\theta }\right\} }^{\mathsf{an}}$ if and only if $\psi _{x}$, evaluation at $x$, annilates the kernel of $\theta $ by Lemma 2.28

Theorem 3.6

Let $S$ and $S'$ be two finitely generated $\mathbb {C}$-algebras and $\phi : S \to S'$ be a $\mathbb {C}$-algebra homomorphism, the natural map ${\left\{ {\operatorname {Spec}{\phi }}\right\} }^{\mathsf{an}} : {\left\{ {\operatorname {Spec}{S’}}\right\} }^{\mathsf{an}} \to {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$ is continuous (in the complex topology) and compatible with the inclusion map, i.e. the following diagram is commutative:

$\begin{tikzcd} [column sep = large] {\an{\specop{S'}}} \arrow{r}{\an{\specop{\phi}}} \arrow[hookrightarrow]{d} & {\an{\specop{S}}} \arrow[hookrightarrow]{d} \\ {\specop{S'}} \arrow{r}{\specop{\phi}} & \specop{S} \end{tikzcd}$

Proof ▶

The commutativity is free. Let us choose generators $\{ a_{1},\dots ,a_{n}\} $’s for $S$ and $\{ b_{1},\dots ,a_{m}\} $’s for $S'$. Let us write the polynomial ring $\mathbb {C}[X_{1},\dots ,X_{n}]$ as $R$ and $\mathbb {C}[Y_{1},\dots , Y_{m}]$ as $R'$. Then we have two surjective $\mathbb {C}$-algebra homomorphisms $\theta : R \to S$ and $\theta ' : R' \to S'$ as usual. Since $\{ b_{i}\} $ generates $S'$, we can find polynomials $P_{i} \in R'$ such that $\phi (a_{i})= P_{i}(b_{1},\dots , b_{m})$. Then we can define a $\mathbb {C}$-algebra homomorphism $\psi : R \to R'$ by $X_{i} \mapsto P_{i}(Y_{1},\dots , Y_{m})$ giving us the following commutative diagrams:

$\begin{tikzcd} [column sep = large] & & & {\complex^{n}} & {\complex^{m}} \\ R \arrow{r}{\psi} \arrow{d}{\theta}& R' \arrow{d}{\theta'} & & {\an{\specop{R}}} \arrow[red]{u}{\sim} & {\an{\specop{R'}}} \arrow{l}[above]{\an{\specop{\psi}}} \arrow[red]{u}{\sim} \\ S \arrow{r}{\phi} & S' & & {\an{\specop{S}}} \arrow[red]{u}{\an{\specop{\theta}}} & {\an{\specop{S'}}} \arrow[red]{u}{\specop{\theta'}} \arrow{l}{\an{\specop{\phi}}} \end{tikzcd}$

The red arrows are continuous because they define the complex topology and ${\left\{ {\operatorname {Spec}{\psi }}\right\} }^{\mathsf{an}}$ is continuous because it is defined by polynomial $\psi $. Thus ${\left\{ {\operatorname {Spec}{\phi }}\right\} }^{\mathsf{an}}$ is continuous.

Corollary 3.7

If $S \to S'$ is an isomorphism of finite $\mathbb {C}$-algebras, then ${\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$ and ${\left\{ {\operatorname {Spec}{S’}}\right\} }^{\mathsf{an}}$ are homeomorphic.

Lemma 3.8

If $\phi : S \to S'$ is a surjective $\mathbb {C}$-algebra homomorphism between two finite $\mathbb {C}$-algebras, then ${\left\{ {\operatorname {Spec}{\phi }}\right\} }^{\mathsf{an}} : {\left\{ {\operatorname {Spec}{S’}}\right\} }^{\mathsf{an}} \to {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$ is an embedding.

Proof ▶

Let $\{ a_{1},\dots , a_{n}\} 's$ be generators of $S$ and $R$ be the polynomial ring $\mathbb {C}[X_{1},\dots , X_{n}]$. Then we have $\theta : R \to S$ such that $\theta (X_{i})=a_{i}$. The composition $R \stackrel{\theta }{\to } S \stackrel{\phi }{\to } S'$ is a surjection as well. Thus by taking ${\left\{ {\operatorname {Spec}(-)}\right\} }^{\mathsf{an}}$ operation, we get

$\begin{tikzcd} [column sep=large] {\an{\specop{S'}}} \arrow{r}{\an{\specop{\phi}}} & {\an{\specop{S}}} \arrow{r}{\an{\specop{\theta}}} & {\an{\specop{R}}} \arrow{r}{\sim} & \complex^n. \end{tikzcd}$

The whole composition is embedding because of independence of generators and ${\left\{ {\operatorname {Spec}{\theta }}\right\} }^{\mathsf{an}}$ is an embedding as well, thus ${\left\{ {\operatorname {Spec}{\phi }}\right\} }^{\mathsf{an}}$ is an embedding as well.

Lemma 3.9

Let us write $\mathbb {C}[X_{1},\dots ,X_{n}]$ as $R$ and let $f \in R$ be a polynomial, then the localization map $\alpha : R \to R_{f}$ induces an embedding ${\left\{ {\operatorname {Spec}{R_{f}}}\right\} }^{\mathsf{an}} \hookrightarrow {\left\{ {\operatorname {Spec}{R}}\right\} }^{\mathsf{an}}$.

Proof ▶

TBD

More generally, we have a corresponding lemma for arbitrary finite $\mathbb {C}$-algebras.

Lemma 3.10

If $S$ is a finite $\mathbb {C}$-algebra and $f \in S$, then the localization map $\alpha : S \to S_{f}$ induces an embedding ${\left\{ {\operatorname {Spec}{S_{f}}}\right\} }^{\mathsf{an}} \hookrightarrow {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$. In fact ${\left\{ {\operatorname {Spec}{S_{f}}}\right\} }^{\mathsf{an}}$ is identified as the subset $D(f) \cap \operatorname {Spec}{S}$ where $D(f)$ is the basic open set in $\operatorname {Spec}{S}$.

Proof ▶

TBD

3.1.2 Arbitrary scheme

Definition 3.11 Complex Topology

Let $\left(X, \mathcal{O}_{X}\right)$ be a scheme locally of finite type over $\mathbb {C}$, let $\mathcal{I}$ be the collection of open immersions $\left(\operatorname *{Spec}{R}, \widetilde{R}\right) \to X$ where $R$ is some finite $\mathbb {C}$-algebra. Then the complex topology on the set of closed points $\max X$ is defined as the weak topology with respect to $\left\{ {\left\{ {{\phi }}\right\} }^{\mathsf{an}} | (\phi , \phi ^{*}) \in \mathcal{I}\right\} $ where ${\left\{ {\phi }\right\} }^{\mathsf{an}}$ is the restriction of $\phi $ to the subset of closed points. When we talk about complex toplogy, we write $\max {X}$ as ${\left\{ {X}\right\} }^{\mathsf{an}}$.

Lemma 3.12

Let $\left(X, \mathcal{O}_{X}\right)$ be a scheme locally of finite type over $\mathbb {C}$, $R$ be a finite $\mathbb {C}$-algebra and $\Psi = (\psi , \psi ^{*}) : \left(\operatorname *{Spec}{R}, \widetilde{R}\right) \to \left(X, \mathcal{O}_{X}\right)$ be an open immersion. Then ${\left\{ {\psi }\right\} }^{\mathsf{an}}$ is an embedding.

Proof ▶

TBD

Lemma 3.13

Let $\lambda _{X} : {\left\{ {X}\right\} }^{\mathsf{an}} \hookrightarrow X$ be the inclusion map, then $\lambda _{X}$ is continuous.

Proof ▶

Let $x \in X$ and $U \cong \operatorname {Spec}A$ be an open affine neighbourhood of $x$. Then we have the following commutative diagram:

$\begin{tikzcd} {\an{\specop{A}}} \arrow[hookrightarrow, red]{d} \arrow{r}{\iota_1} & {\an{X}} \arrow[hookrightarrow]{d}{\lambda_{X}} \\ {\specop{A}} \arrow[hookrightarrow]{r} \arrow{r}{\iota_2} & X. \end{tikzcd}$

The red arrow is continuous by Lemma 3.4; $\iota _{1}$ is continuous by Lemma 3.12; $\iota _{2}$ is an open embedding by hypothesis. Thus $\lambda _{X}$ is continuous as well.

Lemma 3.14

Let $\Phi = (\phi , \phi ^{*}) : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)$ be a morphism of schemes locally of finite type over $\mathbb {C}$, then ${\left\{ {\phi }\right\} }^{\mathsf{an}} : {\left\{ {X}\right\} }^{\mathsf{an}} \to {\left\{ {Y}\right\} }^{\mathsf{an}}$ is continuous.

Proof ▶

Let $x \in X$ and $U \cong \operatorname {Spec}A \subseteq X$ and $V \cong \operatorname {Spec}B \subseteq Y$ such that $\phi U \subseteq V$ be affine neighbourhoods around $x$ and $\phi (x)$ ⁴ . Then $\Phi |_{U} : \left(\operatorname *{Spec}{A}, \widetilde{A}\right) \to \left(\operatorname *{Spec}{B}, \widetilde{B}\right)$ is induced by a $\mathbb {C}$-algebra homomorphism $\alpha : B \to A$, thus we have the following two commutative squares:

$\begin{tikzcd} [column sep=large] {\spec{A}} \arrow{r}{\spec{\alpha}} \arrow[hookrightarrow]{d} & {\spec{B}} \arrow[hookrightarrow]{d} & {\an{\specop{A}}} \arrow[hookrightarrow, red]{d} \arrow{r}{\an{\specop{\alpha}}} & {\an{\specop{B}}} \arrow[hookrightarrow, red]{d}\\ {\schemeOf{X}} \arrow{r}{\Phi} & {\schemeOf{Y}} & {\an{X}} \arrow{r}{\an{\phi}} & \an{Y}. \end{tikzcd}$

where the red arrows are continuous by Lemma 3.13 and ${\left\{ {\operatorname {Spec}\alpha }\right\} }^{\mathsf{an}}$ is continuous by Theorem 3.6. Thus ${\left\{ {\phi }\right\} }^{\mathsf{an}}$ is continuous as well.

Corollary 3.15

Let $\Phi : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)$ be an open immersion of schemes locally of finite type over $\mathbb {C}$, then ${\left\{ {\phi }\right\} }^{\mathsf{an}}$ is an embedding of ${\left\{ {X}\right\} }^{\mathsf{an}}$ into ${\left\{ {Y}\right\} }^{\mathsf{an}}$.

Proof ▶

TBD

Here are two easy consequences:

Corollary 3.16

Given two morphisms among schemes locally of finite type over $\mathbb {C}$

$\begin{tikzcd} {\schemeOf{X}} \arrow{r}{(\phi, \phi^{*})} & {\schemeOf{Y}} \arrow{r}{(\psi, \psi^{*})} & {\schemeOf{Z}}, \end{tikzcd}$

we have ${\left\{ {\psi \circ \phi }\right\} }^{\mathsf{an}} = {\left\{ {\psi }\right\} }^{\mathsf{an}} \circ {\left\{ {\phi }\right\} }^{\mathsf{an}}$

Proof ▶

Restriction of composition is composition of restriction.

Corollary 3.17

Let $\Phi = (\phi , \phi ^{*}) : \left(X, \mathcal{O}_{X}\right) \to \left(Y, \mathcal{O}_{Y}\right)$ be a morphism of schemes locally of finite type over $\mathbb {C}$, we have

$\begin{tikzcd} {\an{X}} \arrow{r}{{\an{\phi}}} \arrow[hookrightarrow]{d} & {\an{Y}} \arrow[hookrightarrow]{d} \\ X \arrow{r}{\phi} & Y \end{tikzcd}$

3.2 Sheaf side

In previous section, we have showed that for any scheme $\left(X, \mathcal{O}_{X}\right)$ locally of finite type over $\mathbb {C}$, we can make a topological space ${\left\{ {X}\right\} }^{\mathsf{an}}$ with the complex topology. In this section, we aim to make a sheaf ${\left\{ {\mathcal{O}_{X}}\right\} }^{\mathsf{an}}$ on ${\left\{ {X}\right\} }^{\mathsf{an}}$. We will consider the affine cases $\left(\operatorname *{Spec}{S}, \widetilde{S}\right)$ where $S$ is a finite $\mathbb {C}$-algebra by choosing generators; then we prove that the construction does not dependent on the choice of generators; then we glue everything together.

3.2.1 Affine schemes

Let $S$ be a finite $\mathbb {C}$-algebra, we choose generators $a_{1},\dots , a_{n}$ and write $\mathbb {C}[X_{1},\dots ,X_{n}]$ as $R$. Thus we have a surjective $\mathbb {C}$-algebra homomorphism $\theta : R \to S$ such that $\theta (X_{i}) = a_{i}$. Then $\ker \theta $ is finitely generated as well, say by $f_{1},\dots ,f_{m}$. Note that the image of ${\left\{ {\operatorname {Spec}\theta }\right\} }^{\mathsf{an}}:{\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}\to {\left\{ {\operatorname {Spec}\mathbb {C}[X_{1},\dots , X_{n}]}\right\} }^{\mathsf{an}}\cong \mathbb {C}^{n}$, which we often just write as ${\left\{ {\operatorname {Spec}\theta }\right\} }^{\mathsf{an}}$ as well, is in bijection with $V(\ker \theta ) := \{ x \in \mathbb {C}^{n}\mid f_{i}(x) = 0~ \text{for all}~ i\} $ ⁵ ; hence is closed in $\mathbb {C}^{n}$ because its the intersection of inverse images of the singleton set $\{ 0\} $.

We do constructions using $f_{1},\dots , f_{m}$, so everything depends on the choice of $a_{1},\dots ,a_{n}$. In this section, we also denote the sheaf of holomorphic function as $\operatorname {\mathcal{Hol}}$ where ${\Gamma \left(\operatorname {\mathcal{Hol}},U\right)} := \{ f : U \to \mathbb {C}| f~ \text{is holomorphic}\} $ for any open subset $U \subseteq \mathbb {C}^{n}$.

We use a specialized basis of topology on $\mathbb {C}^{n}$.

Definition 3.18 Generalized polydiscs

A generalized polydisc

\[ \Delta (g_{1},\dots ,g_{l}; w_{1},\dots , w_{l}; r_{1},\dots , r_{l}) \]

is the set $\left\{ x \in \mathbb {C}^{n}| \left|g_{i}(x) - w_{i}\right| {\lt} r_{i}~ \text{for all}~ i = 1,\dots ,l \right\} $ where each $g_{i} \in \mathbb {C}[X_{1}\dots ,X_{n}]$ is a polynomial of $n$ variables, $w := (w_{1},\dots , w_{n}) \in \mathbb {C}^{l}$ is a point and $r := (r_{1},\dots ,r_{l})\in \mathbb {R}^{l}_{\ge 0}$ are all non-negative. We call $w$ the center of the polydisc and $r$ the polyradius ⁶ .

Remark 3.19

Traditionally, a polydisc $\Delta (w_{1},\dots , w_{l}; r_{1},\dots ,r_{l}) := \{ x \in \mathbb {C}^{l}| \left|w_{i}-r_{i}\right| {\lt} r_{i}\} \} $ is the special case $\Delta (X_{1},\dots ,X_{n};w_{1},\dots ,w_{n}; r_{1},\dots , r_{n})$. For a generalized polydisc $\Delta (g; w; r)$, we have a map $g : \mathbb {C}^{n}\to \mathbb {C}^{l}$ defined by $x \mapsto (g_{1}(x),\dots ,g_{l}(x))$. Note that $\Delta (g; w; r)$ is equal to $g^{-1}\Delta (w; r)$, the inverse image of the usual polydisc. So generalized polydiscs are exactly the inverse image of usual polydiscs by some polynomial map.

Lemma 3.20 Basis of generalized polydiscs

All generalized polydiscs form a topological basis of $\mathbb {C}^{n}$.

Proof ▶

Every open set is a union of generalized polydiscs. Just take small polyradius and $g_{i} = X_{i}$.
Intersection of two generalized polydiscs is a generalized polydisc. Consider $\Delta (g; w; r)$ and $\Delta (g'; w'; r')$. The idea is the following: $\Delta (g; w; r)$ is $g^{-1}\Delta (w; r)$ and $\Delta (g'; w'; r') = {g’}^{-1}\Delta (g'; w; r'))$, and $(g, g'): \mathbb {C}^{n} \to \mathbb {C}^{l}\times \mathbb {C}^{l'}\cong \mathbb {C}^{l + l'}$ is a polynomial map, call it $G$. Since $\Delta (w; r) \times \Delta (w'; r')$ is a usual polydisc, its inverse image under $G$ is a generalized polydiscs and is equal to the intersection of the original generalized polydiscs.

The following terminology is nonstandard.

Definition 3.21 Preanalytification of sheaves.

We have a sheaf $\mathcal{O}^{\mathsf{pre}}$ on $\mathbb {C}^{n}$ is defined to be the cokernel sheaf of the following exact sequence:

$\begin{tikzcd} {\sheafHol^{\oplus m}} \arrow{r} & {\sheafHol} \arrow{r} & {\mathcal{O}^{\mathsf{pre}}} \arrow{r} & 0, \end{tikzcd}$

where the first arrow, on $U$, is given by $(a_{1},\dots ,a_{m}) \mapsto f_{1}|_{U}\symbf {\cdot }a_{1}+\cdots +f_{m}|_{U}\symbf {\cdot }a_{m}$ where $a_{i} \in {\Gamma \left(\operatorname {\mathcal{Hol}},U\right)}$ and $f_{i}$ generates $\ker \theta $

Lemma 3.22

On generalized polydiscs $\Delta _{1} \subseteq \Delta _{2}$, ${\Gamma \left(\mathcal{O}^{\mathsf{pre}},\Delta _{1}\right)}$ is isomorphic to

\[ {{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta _{1}\right)}}\Big/{\ker \theta \symbf {\cdot }{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta _{1}\right)}} \]

and the restriction map ${\Gamma \left({\mathcal{O}^{\mathsf{pre}}},\Delta _{2}\right)} \to {\Gamma \left({\mathcal{O}^{\mathsf{pre}}},\Delta _{1}\right)}$ is the restriction map

\[ \mathrm{res} : {{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta _{2}\right)}}\Big/{\ker \theta \symbf {\cdot }{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta _{2}\right)}} \to {{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta _{1}\right)}}\Big/{\ker \theta \symbf {\cdot }{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta _{1}\right)}} \]

defined by $[f] \mapsto [f\mid _{U}]$.

Proof ▶

This lemma is going to be hard. Uses coherent analytic sheaves, Cartan’s Theorem, Stein manifold, etc. The author of [ 2 , page 108 ] tells us to look at [ 1 , page 136, definition 2; page 243, theorem 2 ] .

Lemma 3.23

Let $x$ be a point not in ${\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$, in another word, $x \in \mathbb {C}^{n} - {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$. There is an open subset $x \in U \subseteq \mathbb {C}^{n}$ such that for any generalized polydiscs $\Delta \subseteq U$, we have ${\Gamma \left(\mathcal{O}^{\mathsf{pre}},\Delta \right)} = 0$.

Proof ▶

By Lemma 2.28, $x$ is not in the image of ${\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$ precisely when at least one of the $f_{i}(x) \ne 0$. Say it’s the first one. Then, there is some open neighbourhood $U$ around $x$ such that $f_{1}\mid _{U}$ is nowhere zero, i.e. invertible. Thus for any generalized polydisc $\Delta \subseteq U$, we have that the ideal $\ker \theta \symbf {\cdot }{\Gamma \left(\operatorname {\mathcal{Hol}},U\right)}$ is everything so that ${\Gamma \left(\mathcal{O}^{pre},\Delta \right)}\cong {{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta \right)}}\Big/{\ker \theta \symbf {\cdot }{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta \right)}}$ is trivial.

Corollary 3.24 $\mathcal{O}^{\mathsf{pre}}$ is supported in ${\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$

Let $V \subseteq \mathbb {C}^{n}-{\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}$ be an open set in $\mathbb {C}^{n}$, then the sections of $\mathcal{O}^{\mathsf{pre}}$ on $V$ is trivial.

Proof ▶

Let $\sigma \in {\Gamma \left(\mathcal{O}^{\mathsf{pre}},V\right)}$, we want to show that $\sigma (x) = 0$ for all $x \in V$. By Lemma 3.23, there exists some open neightbourhood $x \in U \subseteq V$ ⁷ such that sections of $\mathcal{O}^{\mathsf{pre}}$ on any generalized polydiscs contained in $U$ is trivial. Thus we can cover $V$ by a family of generalized polydiscs $\Delta _{i}$ such that $\sigma \mid _{\Delta _{i}}$ are all zero; therefore by sheaf axioms, $\sigma $ is zero.

Corollary 3.25

Let $V\subseteq \mathbb {C}^{n}$ be an open subset then we have an isomorphism

\[ {\Gamma \left(\mathcal{O}^{\mathsf{pre}},V \cup (\mathbb {C}^{n} - {\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}})\right)} \cong {\Gamma \left(\mathcal{O}^{\mathsf{pre}},V\right)} \]

Proof ▶

Write $A := \mathbb {C}^{n}-{\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}$, we need to prove the restriction map from $V \cup A$ to $V$ is both injective and surjective.

Injectivity: suppose a section $\sigma \in {\Gamma \left(\mathcal{O}^{\mathsf{pre}},V \cup A\right)}$ is in the kernel of the restriction map, in another word, $\sigma \mid _{V}$ is zero. Then if we cover $V \cup A$ by $V$ and $A$, we would know that $\sigma \mid _{A}$ is zero as well ⁸ , so by sheaf axiom, $\sigma $ is zero.
Surjectivity: let $\sigma $ be a section in ${\Gamma \left(\mathcal{O}^{\mathsf{pre}},V\right)}$, then we can glue $\sigma $ and $0 \in {\Gamma \left(\mathcal{O}^{\mathsf{pre}},A\right)}$ because $\sigma \mid _{V \cap A}$ must be zero, since ${\Gamma \left(\mathcal{O}^{\mathsf{pre}},V\cap A\right)}$ is trivial by Corollary 3.24.

Definition 3.26 Analytification of sheaf

The analytification ${\left\{ {\widetilde{S}}\right\} }^{\mathsf{an}}$ of $\widetilde{S}$ is a presheaf on ${\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}$ defined by the following:

For any open set $U \subseteq {\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}$, we define the sections ${\Gamma \left({\left\{ {\widetilde{S}}\right\} }^{\mathsf{an}},U\right)}$ to be

\[ {\Gamma \left(\mathcal{O}^{\mathsf{pre}},U \cup \left(\mathbb {C}^{n}-{\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}\right)\right)}. \]
For any open sets $U \subseteq V \subseteq {\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}$, we define the restriction map of ${\left\{ {\widetilde{S}}\right\} }^{\mathsf{an}}$ from $V$ to $U$ is the restriction map of $\mathcal{O}^{\mathsf{pre}}$ from $V \cup \left(\mathbb {C}^{n}-{\left\{ {\operatorname {Spec}S}\right\} }^{\mathsf{an}}\right)$ to $U \cup \left(\mathbb {C}^{n}-{\left\{ {\operatorname {Spec}{S}}\right\} }^{\mathsf{an}}\right)$.

This defines a presheaf, the satisfaction of sheaf axioms are essentially from that of $\mathcal{O}^{\mathsf{pre}}$.

Remark 3.27 Independence of the generators $f_{1},\dots , f_{m}$

Though the definition of $\mathcal{O}^{\mathsf{pre}}$ explicity uses $f_{i}$’s, we see that on the basis of generalized polydiscs, the sections on a generalized polydisc is ${{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta \right)}}\Big/{\ker \theta \symbf {\cdot }{\Gamma \left(\operatorname {\mathcal{Hol}},\Delta \right)}}$ which does not mention any generators. Since two sheaves are isomorphic if their sections on a basis are isomorphic, we must conclude that $\mathcal{O}^{\mathsf{pre}}$ is independent of the choice of $f_{1},\dots ,f_{m}$ as well. But we don’t know yet if the construction is independent from the choice of $a_{1},\dots , a_{m}$.