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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.

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Sounds like you're looking for the spectral sequence relating Cech cohomology and derived functor cohomology: en.wikipedia.org/wiki/… –  Andrew Niles Apr 7 '11 at 6:14
    
Neat! From the description at the bottom, that looks like exactly what I was hoping would exist, though I'm not too familiar with spectral sequences. Why does taking the Cech cohomology of the presheaf that takes $U$ to $H^q(U,\mathcal{F})$ actually related to the Cech cohomology of $\mathcal{F}$? –  Zev Chonoles Apr 7 '11 at 6:19
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Zev: does the following example clarify things? Imagine the cover $(U)$ is just $X$ again! You can easily compute the Cech cohomology of this cover and see that it gives only the global sections of $F$ and no further information about the cohomology of $F$. On $H^0$ you then get an isomorphism (which you always do anyway) and for all the other $H^i$ you get no information at all. Hence the inf of the information we can get from Cech cohomology is "nothing more than global sections". –  Kevin Buzzard Apr 7 '11 at 6:58
    
If a cover is fixed I would not call it Čech cohomology but the cohomology of the cover; the Čech is what one gets by taking a colimit over all (Čech) covers. –  Zoran Skoda Apr 8 '11 at 14:57
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2 Answers

up vote 7 down vote accepted

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.

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So it sounds like the key word here is "spectral sequence", which I've seen once or twice but have not really computed with. At least in theory, would they provide an effective way of computing derived functor cohomology using Cech cohomology on an arbitrary open cover? I guess I was naively hoping just finding the Cech cohomology of $\mathcal{F}$ and the (derived functor) cohomology of $\mathcal{F}$ on the finite intersections would suffice to immediately write down the derived cohomology of $\mathcal{F}$, but we have to go through this gigantic array? –  Zev Chonoles Apr 7 '11 at 6:30
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That naive idea does not work even in the case of two open sets: that is precisely the reason for the M-V long exact sequence. In a way, what you are trying to do is deceptively simple. –  Mariano Suárez-Alvarez Apr 7 '11 at 6:45
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Zev, unfortunately, you'll have to wrestle with this or similar spectral sequence in general. The point of the Leray condition, is that it makes the spectral sequence collapse into something more manageable. –  Donu Arapura Apr 7 '11 at 6:53
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To elaborate a bit further, note that the $E_2$ term of the MV spectral sequence is precisely what you suggested: $$ E_{2}^{p,q} = \bigoplus_{i_0,\dots,i_p} H^q(U_{i_0,\dots,i_p},\mathcal{F}) $$ i.e. the (derived functor) cohomology of the intersections. The higher differentials give the corrections to $H^{n}(\mathcal{F})$ being the direct sum $\oplus_{p+q = n} E_{2}^{p,q}$. This is an excellent starting point to learn about spectral sequences. And I also recommend Bott and Tu. –  Heinrich Hartmann Apr 7 '11 at 16:01
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Ken Brown (of Factorization Lemma fame) published a paper relating Cech hypercohomology and and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:

Click!

Edit: If the link does not work, the paper is Abstract Homotopy Theory and Generalized Sheaf Cohomology by K.S. Brown

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Brown representability is Edgar H. Brown , "Cohomology Theories", Annals of Mathematics Vol 75 no3, (1962) pp 467-484 and "Abstract Homotopy Theory" TAMS (1965), not Ken Brown –  Mike-Doherty Apr 8 '11 at 9:55
    
Oh, oops! I'd better fix that! –  Harry Gindi Apr 8 '11 at 14:24
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