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.
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.
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
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Comment #432 by Herman Rohrbach on
Comment #433 by Johan on