Let me first admit that I know next to nothing about sheaf cohomology, but I might have encountered a good reason to learn it.

Suppose that I have a fibration $F \to E \to B$ and I know that its Leray sheaf is constant with certain coefficients $k$. (For example, this is the case if a $G$ is an elementary abelian $p$-group, $X$ is a $G$-space with a single orbit type, say $H$, and we consider the fibration $EG\times_G X \to X/G$ with fiber $BH$.) I'd like to learn something about cohomology of the total space, so I apply the Serre spectral sequence for singular (or Alexander-Spanier, or Cech) cohomology.

What does the assumption about the Leray sheaf tell me? Is it true that in this case $E_2^{p,q}=H^p(B; H^q(F; k))$ and I don't have to worry about local coefficients?

I seem to recollect that sheaf cohomology with constant sheaf coincides with Cech cohomology, but I am a little bit lost here.

  • 2
    $\begingroup$ If you're on a locally contractible space, then sheaf cohomology of a constant sheaf coincides with singular cohomology. $\endgroup$ – Alex Youcis Aug 14 '15 at 13:52

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