The Stacks project

27.17 Sections over principal opens

Here is a typical result of this kind. We will use a more naive but more direct method of proof in later lemmas.

slogan

Lemma 27.17.1. Let $X$ be a scheme. Let $f \in \Gamma (X, \mathcal{O}_ X)$. Denote $X_ f \subset X$ the open where $f$ is invertible, see Schemes, Lemma 25.6.2. If $X$ is quasi-compact and quasi-separated, the canonical map

\[ \Gamma (X, \mathcal{O}_ X)_ f \longrightarrow \Gamma (X_ f, \mathcal{O}_ X) \]

is an isomorphism. Moreover, if $\mathcal{F}$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules the map

\[ \Gamma (X, \mathcal{F})_ f \longrightarrow \Gamma (X_ f, \mathcal{F}) \]

is an isomorphism.

Proof. Write $R = \Gamma (X, \mathcal{O}_ X)$. Consider the canonical morphism

\[ \varphi : X \longrightarrow \mathop{\mathrm{Spec}}(R) \]

of schemes, see Schemes, Lemma 25.6.4. Then the inverse image of the standard open $D(f)$ on the right hand side is $X_ f$ on the left hand side. Moreover, since $X$ is assumed quasi-compact and quasi-separated the morphism $\varphi $ is quasi-compact and quasi-separated, see Schemes, Lemma 25.19.2 and 25.21.13. Hence by Schemes, Lemma 25.24.1 we see that $\varphi _*\mathcal{F}$ is quasi-coherent. Hence we see that $\varphi _*\mathcal{F} = \widetilde M$ with $M = \Gamma (X, \mathcal{F})$ as an $R$-module. Thus we see that

\[ \Gamma (X_ f, \mathcal{F}) = \Gamma (D(f), \varphi _*\mathcal{F}) = \Gamma (D(f), \widetilde M) = M_ f \]

which is exactly the content of the lemma. The first displayed isomorphism of the lemma follows by taking $\mathcal{F} = \mathcal{O}_ X$. $\square$

Recall that given a scheme $X$, an invertible sheaf $\mathcal{L}$ on $X$, and a sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ we get a graded ring $\Gamma _*(X, \mathcal{L}) = \bigoplus \nolimits _{n \geq 0} \Gamma (X, \mathcal{L}^{\otimes n})$ and a graded $\Gamma _*(X, \mathcal{L})$-module $\Gamma _*(X, \mathcal{L}, \mathcal{F}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \Gamma (X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n})$ see Modules, Definition 17.22.7. If we have moreover a section $s \in \Gamma (X, \mathcal{L})$, then we obtain a map

27.17.1.1
\begin{equation} \label{properties-equation-module-invert-s} \Gamma _*(X, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow \Gamma (X_ s, \mathcal{F}|_{X_ s}) \end{equation}

which sends $t/s^ n$ where $t \in \Gamma (X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n})$ to $t|_{X_ s} \otimes s|_{X_ s}^{-n}$. This makes sense because $X_ s \subset X$ is by definition the open over which $s$ has an inverse, see Modules, Lemma 17.22.10.

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.

  1. If $X$ is quasi-compact, then (27.17.1.1) is injective, and

  2. 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 25.21.6. 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. $\square$

Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $\mathcal{F}$ and $\mathcal{G}$ be quasi-coherent $\mathcal{O}_ X$-modules. Consider the graded $\Gamma _*(X, \mathcal{L})$-module

\[ M = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) \]

Next, let $s \in \Gamma (X, \mathcal{L})$ be a section. Then there is a canonical map

27.17.2.1
\begin{equation} \label{properties-equation-hom-invert-s} M_{(s)} \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ s}}(\mathcal{F}|_{X_ s}, \mathcal{G}|_{X_ s}) \end{equation}

which sends $\alpha /s^ n$ to the map $\alpha |_{X_ s} \otimes s|_{X_ s}^{-n}$. The following lemma, combined with Lemma 27.22.4, says roughly that, if $X$ is quasi-compact and quasi-separated, the category of finitely presented $\mathcal{O}_{X_ s}$-modules is the category of finitely presented $\mathcal{O}_ X$-modules with the multiplicative system of maps $s^ n: \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ inverted.

Lemma 27.17.3. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$ be a section. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_ X$-modules.

  1. If $X$ is quasi-compact and $\mathcal{F}$ is of finite type, then (27.17.2.1) is injective, and

  2. if $X$ is quasi-compact and quasi-separated and $\mathcal{F}$ is of finite presentation, then (27.17.2.1) is bijective.

Proof. We first prove the lemma in case $X = \mathop{\mathrm{Spec}}(A)$ is affine and $\mathcal{L} = \mathcal{O}_ X$. In this case $s$ corresponds to an element $f \in A$. Say $\mathcal{F} = \widetilde{M}$ and $\mathcal{G} = \widetilde{N}$ for some $A$-modules $M$ and $N$. Then the lemma translates (via Lemmas 27.16.1 and 27.16.2) into the following algebra statements

  1. If $M$ is a finite $A$-module and $\varphi : M \to N$ is an $A$-module map such that the induced map $M_ f \to N_ f$ is zero, then $f^ n\varphi = 0$ for some $n$.

  2. If $M$ is a finitely presented $A$-module, then $\mathop{\mathrm{Hom}}\nolimits _ A(M, N)_ f = \mathop{\mathrm{Hom}}\nolimits _{A_ f}(M_ f, N_ f)$.

The second statement is Algebra, Lemma 10.10.2 and we omit the proof of the first statement.

Next, we prove (1) for general $X$. Assume $X$ is quasi-compact and hoose 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 $X_ s \cap U_ j = D(f_ j)$. Let $\alpha /s^ n$ be an element in the kernel of (27.17.2.1). Then $\alpha |_{X_ s} = 0$. Hence $(\alpha |_{U_ j})|_{D(f_ j)} = 0$. By the affine case treated above we conclude that $f_ j^{e_ j} \alpha |_{U_ j} = 0$ for some $e_ j \geq 0$. Let $e = \max (e_ j)$. Then we see that $\alpha \otimes s^ e$ restricts to zero on $U_ j$ for all $j$, hence is zero. Since $\alpha /s^ n$ is equal to $\alpha \otimes s^ e/s^{n + e}$ in $M_{(s)}$ we conclude that $\alpha /s^ n = 0$ as desired.

Proof of (2). Since $\mathcal{F}$ is of finite presentation, the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent, see Schemes, Section 25.24. Moreover, it is clear that

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n} \]

for all $n$. Hence in this case the statement follows from Lemma 27.17.2 applied to $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$. $\square$


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