Let $X$ be a paracompact topological space or a manifold (which is not a particular case since the structure sheaves are different). It is well-known that vector bundles (more generally, $\mathcal{O}\_X$-modules) are soft, hence acyclic for $\Gamma$.


**I wonder if the same holds for $\Gamma\_c$.**



Are vector bundles acyclic for $\Gamma_c$?
Yes, at least if $X$ is locally compact.


A useful reference is Bredon's book. In brief, soft sheaves are in particular $c$-soft (they satisfy the softness condition for *compact* subsets of $X$ as opposed to all closed subsets of $X$). On a locally compact space, $c$-soft sheaves are $\Gamma\_c$-acyclic. Bredon states a more general version of this for an arbitrary "family of supports" $\Phi$, where "locally compact" becomes "the family of supports $\Phi$ is paracompactifying".


