Let $X$ be a topological space and $U$ an open cover of $X$.
In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:
The zeroth Cech cohomology functor $\tilde H^0(U,-):Pre(X)\to Ab$ from presheaves on $X$ to abelian groups is left exact and its right derived functors coincide with the cohomology $\tilde H^n(U,-)$ of the chech complex. So one may interpret Cech cohomology as derived presheaf cohomology.
On the other hand the inclusion $i:Shv(X)\to Pre(X)$ of sheaves on $X$ into presheaves is left exact and the diagram [ \begin{array}{rcl} Shv(X)&\xrightarrow{\Gamma_X}&Ab\\ i\searrow&&\nearrow \tilde H^0(U,-)\\ &Pre(X)& \end{array} ] commutes. Let $F\in Shv(X)$ be a sheaf. The derived functors $H^n(X,F)$ of the left exact functor $\Gamma_X$ are called sheaf cohomology. They are in general not equal to the derived functors $\tilde H^n(U,-)$ (Cech cohomology). The relation between the two is the spectral sequence
$$ E_2^{p,q}=\tilde H^p(U,H^q(-,F))\Rightarrow H^{p+q}(X,F) $$
induced by the Grothendieck spectral sequence.
- What is the general picture behind this?
- In this thread it is explained why the presheaf $H^q(-,F)$ in the spectral sequence sheafifies to zero for $q\geq 1$. How can one interpret this?
- For $X$ locally contractible and $F$ the sheaf of localy constant functions, sheaf cohomology equals singular cohomology and for a cover $U$ with two open sets the spectral sequence is just the Mayer Vietoris sequence. How does this fit into the general picture?
