## Tag `01PW`

Chapter 27: Properties of Schemes > Section 27.17: Sections over principal opens

Lemma 27.17.2. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $X$. Let $s \in \Gamma(X, \mathcal{L})$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.

- If $X$ is quasi-compact, then (27.17.1.1) is injective, and
- if $X$ is quasi-compact and quasi-separated, then (27.17.1.1) is an isomorphism.
In particular, the canonical map $$ \Gamma_*(X, \mathcal{L})_{(s)} \longrightarrow \Gamma(X_s, \mathcal{O}_X),\quad a/s^n \longmapsto a \otimes s^{-n} $$ is an isomorphism if $X$ is quasi-compact and quasi-separated.

Proof.Assume $X$ is quasi-compact. Choose a finite affine open covering $X = U_1 \cup \ldots \cup U_m$ with $U_j$ affine and $\mathcal{L}|_{U_j} \cong \mathcal{O}_{U_j}$. Via this isomorphism, the image $s|_{U_j}$ corresponds to some $f_j \in \Gamma(U_j, \mathcal{O}_{U_j})$. Then $X_s \cap U_j = D(f_j)$.Proof of (1). Let $t/s^n$ be an element in the kernel of (27.17.1.1). Then $t|_{X_s} = 0$. Hence $(t|_{U_j})|_{D(f_j)} = 0$. By Lemma 27.17.1 we conclude that $f_j^{e_j} t|_{U_j} = 0$ for some $e_j \geq 0$. Let $e = \max(e_j)$. Then we see that $t \otimes s^e$ restricts to zero on $U_j$ for all $j$, hence is zero. Since $t/s^n$ is equal to $t \otimes s^e/s^{n + e}$ in $\Gamma_*(X, \mathcal{L}, \mathcal{F})_{(s)}$ we conclude that $t/s^n = 0$ as desired.

Proof of (2). Assume $X$ is quasi-compact and quasi-separated. Then $U_j \cap U_{j'}$ is quasi-compact for all pairs $j, j'$, see Schemes, Lemma 25.21.7. By part (1) we know (27.17.1.1) is injective. Let $t' \in \Gamma(X_s, \mathcal{F}|_{X_s})$. For every $j$, there exist an integer $n_j \geq 0$ and $t'_j \in \Gamma(U_j, \mathcal{F}|_{U_j})$ such that $t'|_{D(f_j)}$ corresponds to $t'_j/f_j^{e_j}$ via the isomorphism of Lemma 27.17.1. Set $e = \max(e_j)$ and $$ t_j = t'_j \otimes s|_{U_j}^e \in \Gamma(U_j, (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes e})|_{U_j}) $$ Then we see that $t_j|_{U_j \cap U_{j'}}$ and $t_{j'}|_{U_j \cap U_{j'}}$ map to the same section of $\mathcal{F}$ over $U_j \cap U_{j'} \cap X_s$. By quasi-compactness of $U_j \cap U_{j'}$ and part (1) there exists an integer $e' \geq 0$ such that $$ t_j|_{U_j \cap U_{j'}} \otimes s^{e'}|_{U_j \cap U_{j'}} = t_{j'}|_{U_j \cap U_{j'}} \otimes s^{e'}|_{U_j \cap U_{j'}} $$ as sections of $\mathcal{F} \otimes \mathcal{L}^{\otimes e + e'}$ over $U_j \cap U_{j'}$. We may choose the same $e'$ to work for all pairs $j, j'$. Then the sheaf conditions implies there is a section $t \in \Gamma(X, \mathcal{F} \otimes \mathcal{L}^{\otimes e + e'})$ whose restriction to $U_j$ is $t_j \otimes s^{e'}|_{U_j}$. A simple computation shows that $t/s^{e + e'}$ maps to $t'$ as desired. $\square$

The code snippet corresponding to this tag is a part of the file `properties.tex` and is located in lines 2003–2022 (see updates for more information).

```
\begin{lemma}
\label{lemma-invert-s-sections}
Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $X$.
Let $s \in \Gamma(X, \mathcal{L})$. Let $\mathcal{F}$ be a quasi-coherent
$\mathcal{O}_X$-module.
\begin{enumerate}
\item If $X$ is quasi-compact, then (\ref{equation-module-invert-s})
is injective, and
\item if $X$ is quasi-compact and quasi-separated, then
(\ref{equation-module-invert-s}) is an isomorphism.
\end{enumerate}
In particular, the canonical map
$$
\Gamma_*(X, \mathcal{L})_{(s)}
\longrightarrow
\Gamma(X_s, \mathcal{O}_X),\quad
a/s^n \longmapsto a \otimes s^{-n}
$$
is an isomorphism if $X$ is quasi-compact and quasi-separated.
\end{lemma}
\begin{proof}
Assume $X$ is quasi-compact. Choose a finite affine open covering
$X = U_1 \cup \ldots \cup U_m$ with $U_j$ affine and
$\mathcal{L}|_{U_j} \cong \mathcal{O}_{U_j}$. Via this isomorphism,
the image $s|_{U_j}$ corresponds to some
$f_j \in \Gamma(U_j, \mathcal{O}_{U_j})$. Then
$X_s \cap U_j = D(f_j)$.
\medskip\noindent
Proof of (1). Let $t/s^n$ be an element in the kernel of
(\ref{equation-module-invert-s}). Then $t|_{X_s} = 0$.
Hence $(t|_{U_j})|_{D(f_j)} = 0$. By
Lemma \ref{lemma-invert-f-sections} we conclude that
$f_j^{e_j} t|_{U_j} = 0$ for some
$e_j \geq 0$. Let $e = \max(e_j)$. Then we see that $t \otimes s^e$
restricts to zero on $U_j$ for all $j$, hence is zero. Since $t/s^n$
is equal to $t \otimes s^e/s^{n + e}$ in
$\Gamma_*(X, \mathcal{L}, \mathcal{F})_{(s)}$ we conclude that $t/s^n = 0$
as desired.
\medskip\noindent
Proof of (2). Assume $X$ is quasi-compact and quasi-separated.
Then $U_j \cap U_{j'}$ is quasi-compact for all pairs $j, j'$, see
Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated}.
By part (1) we know (\ref{equation-module-invert-s}) is injective.
Let $t' \in \Gamma(X_s, \mathcal{F}|_{X_s})$. For every $j$, there exist an
integer $n_j \geq 0$ and $t'_j \in \Gamma(U_j, \mathcal{F}|_{U_j})$ such that
$t'|_{D(f_j)}$ corresponds to $t'_j/f_j^{e_j}$
via the isomorphism of Lemma \ref{lemma-invert-f-sections}.
Set $e = \max(e_j)$ and
$$
t_j = t'_j \otimes s|_{U_j}^e \in
\Gamma(U_j,
(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes e})|_{U_j})
$$
Then we see that $t_j|_{U_j \cap U_{j'}}$ and $t_{j'}|_{U_j \cap U_{j'}}$
map to the same section of $\mathcal{F}$ over $U_j \cap U_{j'} \cap X_s$.
By quasi-compactness of $U_j \cap U_{j'}$ and part (1) there exists an
integer $e' \geq 0$ such that
$$
t_j|_{U_j \cap U_{j'}} \otimes s^{e'}|_{U_j \cap U_{j'}} =
t_{j'}|_{U_j \cap U_{j'}} \otimes s^{e'}|_{U_j \cap U_{j'}}
$$
as sections of $\mathcal{F} \otimes \mathcal{L}^{\otimes e + e'}$ over
$U_j \cap U_{j'}$. We may choose the same $e'$ to work for all pairs
$j, j'$. Then the sheaf conditions implies there is a section
$t \in \Gamma(X, \mathcal{F} \otimes \mathcal{L}^{\otimes e + e'})$
whose restriction to $U_j$ is $t_j \otimes s^{e'}|_{U_j}$.
A simple computation shows that $t/s^{e + e'}$ maps to $t'$
as desired.
\end{proof}
```

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