In my algebraic geometry class this semester, we've learned about Leray's Theorem, which states that for a sheaf $\mathcal{F}$ on a topological space $X$, and $\mathcal{U}$ a countable cover of $X$, if $\mathcal{F}$ is acyclic on every finite intersection of elements of $\mathcal{U}$ then the Cech cohomology $\check{H}^p(\mathcal{U},\mathcal{F})$ and derived functor cohomology $H^p(X,\mathcal{F})$ agree.
The potential for disagreement between them is covered well in these two MO questions. However, what neither of them seem to address is whether we can salvage any information about $H^p(X,\mathcal{F})$ from $\check{H}^p(\mathcal{U},\mathcal{F})$ even when $\mathcal{U}$ does not have the property that $\mathcal{F}$ is acyclic on all finite intersections, which is what I'd like to find out about here. I'm aware of Hartshorne Lemma 3.4.4, which says that there is a natural map $\check{H}^p(\mathcal{U},\mathcal{F})\rightarrow H^p(X,\mathcal{F})$ which is functorial in $\mathcal{F}$, but this is gotten by abstract nonsense – my feeling is that the existence of this map is not conveying much useful information. For all we know (?), all these maps could be the trivial homomorphism.
What I'm imagining is that perhaps the higher cohomology of $\mathcal{F}$ on the finite intersections of $\mathcal{U}$ can be related to the "difference" between $\check{H}^p(\mathcal{U},\mathcal{F})$ and $H^p(X,\mathcal{F})$, and that when the higher cohomology vanishes (i.e. $\mathcal{F}$ is acyclic), we get back the original theorem (that Cech and derived functor agree).
So, is there a useful relationship betwen Cech and derived functor cohomology even when $\mathcal{U}$ is not a nice open cover with respect to $\mathcal{F}$? Am I mistaken in assuming that the map $\check{H}^p(\mathcal{U},\mathcal{F})\rightarrow H^p(X,\mathcal{F})$ is not (particularly) useful?
Also I would like to avoid if possible the operation of taking the limit over all covers of $X$. I want to relate the specific Cech cohomology with respect to the cover $\mathcal{U}$, whatever its failings may be, with the derived functor cohomology.
Best Answer
There is a Mayer-Vietoris spectral sequence relating the two. This is a "direct" generalization of the Mayer-Vietoris long exact sequence, which is the special case in which your covering has just two open sets.
This is explained, if I recall correctly, in Bott-Tu.