\(\mcf\) is flasque, or flabby, if every restriction \(\mcf(U) \to \mcf(V)\), for \(V \subseteq U\), is surjective.
Injective implies flasque, flasque implies \(\globsec{X;\wait}\dash\)acyclic, and for a short exact sequence with flasque left term the sequence of global sections stays exact.
Every sheaf \(\mcf\) embeds in the flasque sheaf \(\prod_x j^x_* \mcf_x\) of discontinuous sections, so every sheaf has a resolution by flasque, hence \(\globsec{X;\wait}\dash\)acyclic, sheaves.
The proof of acyclicity is induction on the sequence \(0 \to \mcf \to \mci \to \mcg \to 0\) with \(\mci\) injective: flasqueness of \(\mcf\) forces \(\mcg\) flasque and the global sections exact, so \(H^1(\mcf) = 0\) and the higher groups shift down.
(Flasque sheaves)
The following sheaves are flasque:
the sheaf \(\prod_x j^x_* \mcf_x\) of discontinuous sections of any sheaf \(\mcf\), where \(j^x\colon\ts{x}\injects X\);
every constant sheaf on an irreducible topological space [@Har10a, Exercise II.1.16];
the sheaf \(\tilde{I}\) on \(\Spec A\) for a Noetherian ring \(A\) and an injective \(A\dash\)module \(I\) [@Har10a, Proposition III.3.4]; Hartshorne uses this to prove that \(H^i(\Spec A,\tilde M)=0\) for \(i>0\).
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D-COHDERSheaf cohomology as the derived functor of global sectionsuses
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D-SHFFINESoft and fine sheavesrelated-toT-COHAFFHigher cohomology vanishes on an affine schemeusesD-COHGODEMENTThe Godement resolutionuses