We shall now approach the proof of the Cartan vanishing theorem. First, however, it will be necessary to describe a spectral sequence between Cech cohomology and derived functor cohomology. For now, the reason is that there isn’t any obvious way for us to compute derived functor cohomology, because injective sheaves are big and scary, while Cech cohomology is nice and concrete. And indeed, all we’ve done so far is compute various Cech cohomologies.

I should mention that I don’t know a standard reference for the material in this post. I didn’t find Godement’s treatment in Theorie des faisceaux to be terribly enlightening, but after a fair bit of googling I found a sketch in James Milne’s online notes on étale cohomology. Fortunately, enough details are given to enable one to work it out more fully for oneself.

Let ${X}$ be a topological space covered by an open cover ${\mathfrak{A} = \left\{U_i\right\}_{i \in I}}$, and consider the category ${\mathfrak{C}}$ of presheaves of abelian groups on ${X}$. Let ${\mathfrak{C}'}$ be the subcategory of sheaves. The spectral sequence will be the Grothendieck spectral sequence of the composite of functors $\displaystyle \mathfrak{C}' \stackrel{F}{\rightarrow} \mathfrak{C} \stackrel{G}{\rightarrow} \mathbf{Ab}.$

Here ${F}$ is the inclusion of the subcategory, and ${G}$ is the functor sending a presheaf to its zeroth Cech cohomology. The first thing to note is that the functor $\displaystyle \mathcal{F} \rightarrow \mathcal{F}(U), \mathfrak{C} \rightarrow \mathbf{Ab}$

is an exact functor as we are working with presheaves. For presheaves, exactness can be checked on open sets instead of stalks. Let ${\mathfrak{D}}$ be the category of abelian groups. We have a functor that sends a presheaf ${\mathcal{F}}$ to its zeroth Cech cohomology.

One ought to note that Cech cohomology makes sense in a presheaf. To recall what this means, note that the cochains in dimension ${r-1}$ are the same thing as alternating maps ${\phi}$ out of ${I^r}$ such that ${\phi(i_1, \dots, i_r)}$ takes values in ${\mathcal{F}(U_{i_1} \cap \dots U_{i_r})}$. The coboundary map is the usual: ${\partial \phi(i_1, \dots, i_{r+1}) = \sum (-1)^j \phi(i_1, \dots, \hat{i_j}, \dots, i_{r+1}).}$ For a sheaf, the zeroth Cech cohomology is—as is easy to check—the space of global sections. This is not necessarily true for a presheaf, because the proof for sheaves uses the glueability of sections. To compute the spectral sequence, we will have to find the derived functors of ${F,G}$. This will take a bit of checking.

Proposition 38 The ${i}$th derived functor of ${F}$ sends a sheaf ${\mathcal{F}}$ into the presheaf ${\mathcal{H}^i(\mathcal{F}) = \left\{U \rightarrow H^i(U, \mathcal{F})\right\}}$.

Proof: The functor as described is a ${\delta}$-functor from sheaves to presheaves. Indeed, this follows from the fact that for a short exact sequence of sheaves $\displaystyle 0 \rightarrow \mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0,$

there is an associated long exact sequence, for each open set ${U}$, $\displaystyle H^i(U,\mathcal{F}' ) \rightarrow H^i(U, \mathcal{F}) \rightarrow H^i(U, \mathcal{F}'') \rightarrow H^{i+1}(\mathcal{F}', U) \rightarrow \dots.$

Since exactness of presheaves is equivalent to exactness of the sections over each open set, we find that $\displaystyle \mathcal{H}^i(\mathcal{F}') \rightarrow \mathcal{H}^i(\mathcal{F}) \rightarrow \mathcal{H}^i(\mathcal{F}'') \rightarrow \mathcal{H}^{i+1}(\mathcal{F}') \rightarrow \dots.$

I claim that this is an effaceable ${\delta}$-functor. It isn’t that important to know the precise definition (I’m pretty sure it’s in Grothendieck’s Tohoku paper), but the point is that it vanishes on injectives. Then, from the universal property of derived functors, and the fact that ${\mathcal{H}^0}$ is the inclusion functor ${F}$, it will follow that the ${i}$th derived functor of ${F}$ is ${\mathcal{H}^i}$.

But if ${\mathcal{I}}$ is an injective sheaf, then it is injective over every open set, so ${H^i(U, \mathcal{I})=0}$ for ${i >0}$. (Alternatively, this follows because an injective sheaf is flabby.) In particular, ${\mathcal{H}^i(\mathcal{I})=0}$ for ${i>0}$. So these are the derived functors of ${F}$. So we know what the derived functors of ${F}$ look like.

Now, we need to get a picture of the derived functors of ${G}$. We will show that these are just the higher Cech cohomologies. I should add a caveat that this is not true for the category of sheaves! It is important here that we are working for ${G}$ as a functor on the category of presheaves. On the category of sheaves, the Cech functors don’t generally form a ${\delta}$-functor. The derived functors of the zeroth Cech functor ${H^0(\mathfrak{A}, -)}$ are just the usual cohomologies because ${H^0(\mathfrak{A}, -)}$ is the set of global sections.

Proposition 39 The derived functors of ${G}$ are the functors ${H^i(\mathfrak{A}, -)}$ on the category of sheaves.

Proof: The first thing to check is that the ${H^i(\mathfrak{A}, -)}$ form a ${\delta}$-functor. Again, this is all because we are on the category of presheaves. So say $\displaystyle 0 \rightarrow \mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0$

is an exact sequence of presheaves. Then $\displaystyle 0 \rightarrow \mathcal{F'}(U) \rightarrow \mathcal{F}(U) \rightarrow \mathcal{F}''(U) \rightarrow 0$

is exact. So in particular by taking products of this, we get an exact sequence of Cech complexes. Thus taking cohomology, we get a natural long exact sequence in the Cech cohomology.

Finally, we need to check that the ${H^i(\mathfrak{A}, -)}$ form a universal ${\delta}$-functor. In particular, by the Tohoku nonsense, we need to show that these vanish (for ${i>0}$) on injective objects in the category of presheaves. This takes a little work, but it’s done in Tamme’s “Introduction to Etale Cohomology.” (Removed an incorrect argument here.)

All right. We’re almost there. We have the two functors ${F, G }$ between fairly nice abelian categories, and we have computed the derived functors of each of them. The composite ${G \circ F}$ is the global section functor on the category of sheaves, and its derived functors are the usual sheaf cohomology. So we will be able to write down a spectral sequence to compute the usual sheaf cohomology. But before that, we have to check a technical condition that one needs before applying the Grothendieck spectral sequence. Fortunately, it is fairly easy.

Proposition 40 ${F}$ sends injectives into ${G}$-acyclics.

Proof: An injective sheaf is flabby, and a flabby presheaf has trivial Cech cohomology, as we have seen. In particular, this means that the Grothendieck spectral sequence applies.

Theorem 41 There is a convergent spectral sequence whose ${E^2}$ page is $\displaystyle {H}^p(\mathfrak{A}, \mathcal{H}^q(\mathcal{F})) \rightarrow H^{p+q}(X, \mathcal{F}).$

Here ${\mathcal{H}^q(\mathcal{F})}$ is the presheaf ${U \rightarrow H^q(U, \mathcal{F})}$.

This is now immediate from the Grothendieck spectral sequence, because, if ${R}$ denotes the operator of taking a derived functor, we have seen: $\displaystyle R^{p+q}(G \circ F) = H^{p+q}(X, -)$

and $\displaystyle R^p(F) = H^p(\mathfrak{A}, -), \quad R^q(G) = \mathcal{H}^q(-).$