Call a sheaf flasque if for all open sets $U \subset V$, the restriction map$$\mathcal{F}(V) \to \mathcal{F}(U)$$is surjective. Is every sheaf a subsheaf of a flasque sheaf?

  • $\begingroup$ Yes, do you know what the étalé space of a sheaf is? $\endgroup$ – user1971 Oct 10 '15 at 19:40
  • 2
    $\begingroup$ The search phrase here is "Godement resolution". $\endgroup$ – Hoot Oct 10 '15 at 21:31
  • 1
    $\begingroup$ or 'sheaf of discontinuous sections.' $\endgroup$ – Tomo Nov 7 '15 at 4:29

Let$$\mathcal{G} = \prod_{p \in X} \mathcal{F}_p.$$Then $\mathcal{F}$ is a subsheaf of $\mathcal{G}$ and $\mathcal{G}$ is easily seen to be flasque.

  • $\begingroup$ It may be a simple question, but how can you prove that $\mathcal{F}$ is a subsheaf of $\mathcal{G}$? I'm new to sheaf theory and I need to prove that there exists a sheaf monomorphism from any sheaf $\mathcal{G}$ to a flasque sheaf, which I expect to be this $\mathcal{G}$. If so, I guess it is equivalent to your statement. I could open a new question but it has to risk to be quite similar to this one, so... Thanks for your help :) $\endgroup$ – MoebiusCorzer Jan 18 '16 at 16:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy