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 $U$ of $x$ 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 " open subspace of ...", which should be " open subspace of ..."

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