\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

17.8 Modules locally generated by sections

Let $(X, \mathcal{O}_ X)$ be a ringed space. In this and the following section we will often restrict sheaves to open subspaces $U \subset X$, see Sheaves, Section 6.31. In particular, we will often denote the open subspace by $(U, \mathcal{O}_ U)$ instead of the more correct notation $(U, \mathcal{O}_ X|_ U)$, see Sheaves, Definition 6.31.2.

Consider the open immersion $j : U = (0 , \infty ) \to \mathbf{R} = X$, and the abelian sheaf $j_!\underline{\mathbf{Z}}_ U$. By Sheaves, Section 6.31 the stalk of $j_!\underline{\mathbf{Z}}_ U$ at $x = 0$ is $0$. In fact the sections of this sheaf over any open interval containing $0$ are $0$. Thus there is no open neighbourhood of the point $0$ over which the sheaf can be generated by sections.

Definition 17.8.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We say that $\mathcal{F}$ is locally generated by sections if for every $x \in X$ there exists an open neighbourhood $U$ such that $\mathcal{F}|_ U$ is globally generated as a sheaf of $\mathcal{O}_ U$-modules.

In other words there exists a set $I$ and for each $i$ a section $s_ i \in \mathcal{F}(U)$ such that the associated map

\[ \bigoplus \nolimits _{i \in I} \mathcal{O}_ U \longrightarrow \mathcal{F}|_ U \]

is surjective.

Lemma 17.8.2. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ is locally generated by sections if $\mathcal{G}$ is locally generated by sections.

Proof. Given an open subspace $V$ of $Y$ we may consider the commutative diagram of ringed spaces

\[ \xymatrix{ (f^{-1}V, \mathcal{O}_{f^{-1}V}) \ar[r]_{j'} \ar[d]_{f'} & (X, \mathcal{O}_ X) \ar[d]^ f \\ (V, \mathcal{O}_ V) \ar[r]^ j & (Y, \mathcal{O}_ Y) } \]

We know that $f^*\mathcal{G}|_{f^{-1}V} \cong (f')^*(\mathcal{G}|_ V)$, see Sheaves, Lemma 6.26.3. Thus we may assume that $\mathcal{G}$ is globally generated.

We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma 17.3.3. Thus if we have a surjection

\[ \bigoplus \nolimits _{i \in I} \mathcal{O}_ Y \to \mathcal{G} \to 0 \]

then upon applying $f^*$ we obtain the surjection

\[ \bigoplus \nolimits _{i \in I} \mathcal{O}_ X \to f^*\mathcal{G} \to 0. \]

This implies the lemma. $\square$


Comments (2)

Comment #432 by Herman Rohrbach on

There is a small error (typo?) in the first line of the proof of lemma 17.8.2; it says "...an open subspace of ...", which should be "...an open subspace of ..."


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01B1. Beware of the difference between the letter 'O' and the digit '0'.