Flasque Sheaves 2

[Tsakanikas Nickos]

Let \( X \) be a topological space and let
\[ 0 \longrightarrow \mathcal{F} \longrightarrow \mathcal{G} \longrightarrow \mathcal{H} \longrightarrow 0 \]be a short exact sequence of sheaves of abelian groups on \( X \). First, show that if \( \mathcal{F} \) is flasque, then for any open subset \( U \) of \( X \) the sequence
\[ 0 \longrightarrow \mathcal{F}(U) \longrightarrow \mathcal{G}(U) \longrightarrow \mathcal{H}(U) \longrightarrow 0 \]of abelian groups is exact. Second, (using the previous result) show that if additionally \( \mathcal{G} \) is flasque, then \( \mathcal{H} \) is also flasque.