# The Stacks Project

## Tag 01B1

### 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. $\square$

The code snippet corresponding to this tag is a part of the file modules.tex and is located in lines 710–811 (see updates for more information).

\section{Modules locally generated by sections}
\label{section-locally-generated}

\noindent
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 \ref{sheaves-section-open-immersions}.
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 \ref{sheaves-definition-restriction}.

\medskip\noindent
Consider the open immersion
$j : U = (0 , \infty) \to \mathbf{R} = X$, and the abelian sheaf
$j_!\underline{\mathbf{Z}}_U$. By Sheaves, Section
\ref{sheaves-section-open-immersions} 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.

\begin{definition}
\label{definition-locally-generated}
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 {\it 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.
\end{definition}

\noindent
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.

\begin{lemma}
\label{lemma-pullback-locally-generated}
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.
\end{lemma}

\begin{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 \ref{sheaves-lemma-push-pull-composition-modules}.
Thus we may assume that $\mathcal{G}$ is globally generated.

\medskip\noindent
We have seen that $f^*$ commutes with all colimits,
and is right exact, see Lemma \ref{lemma-exactness-pushforward-pullback}.
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.
\end{proof}

Comment #432 by Herman Rohrbach on January 25, 2014 a 3:13 pm UTC

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

Comment #433 by Johan (site) on January 25, 2014 a 11:43 pm UTC

Fixed here. Thanks!

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