## 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 sectionsif 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}
```

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