The next big application of the Koszul complex and this general machinery that I have in mind is to projective space. Namely, consider a ring ${A}$, and an integer ${n \in \mathbb{Z}_{\geq 0}}$. We have the ${A}$-scheme ${\mathop{\mathbb P}^n_A = \mathrm{Proj} A[x_0, \dots, x_n]}$. Recall that on it, we have canonical line bundles ${\mathcal{O}(m)}$ for each ${m \in \mathbb{Z}}$, which come from homogeneous localization of the ${A[x_0, \dots, x_n]}$-modules obtained from ${A[x_0, \dots, x_n]}$ itself by twisting the degrees by ${m}$. When ${A}$ is a field, the only line bundles on it are of this form. (I am not sure if this is true in general. I think it will be true, but perhaps someone can confirm.)

It will be useful to compute the cohomology of these line bundles. For one thing, this will lead to Serre duality, from a very convenient isomorphism that will spring up. For another, we will see that they are finitely generated over ${A}$. This is far from obvious. The scheme ${\mathop{\mathbb P}^n_A}$ is not finite over ${A}$, and a priori this is not expected.

But to start, let’s think more abstractly. Let ${X}$ be any quasi-compact, quasi-separated scheme; we’ll assume this for reasons below. Let ${\mathcal{L}}$ be a line bundle on ${X}$, and ${\mathcal{F}}$ an arbitrary quasi-coherent sheaf. We can consider the twists ${\mathcal{F} \otimes \mathcal{L}^{\otimes m}}$ for any ${m \in \mathbb{Z}}$. This is a bunch of sheaves, but it is something more.

Let us package these sheaves together. Namely, let us consider the sheaves:

$\displaystyle \bigoplus \mathcal{L}^{\otimes m}, \quad \mathcal{H}=\bigoplus \mathcal{F} \otimes \mathcal{L}^{\otimes m}$

Earlier, our proof of the vanishing of higher quasi-coherent cohomology on an affine was actually very incomplete. We actually computed only Cech cohomology, and waved our hands while pointing to a fancy sheaf-theoretic result of Cartan. I would like to prove this result today, following Godement’s Theorie des faisceaux.

Cech cohomology is (comparatively) easy to compute, for instance via the Koszul complex. But the problem is that we don’t a priori know if coincides with derived functor cohomology. We have a natural map between Cech and derived functor cohomology in any case, but in general it won’t be an isomorphism. Leray’s theorem is a sufficient condition for this, but its expression is fundamentally in terms of derived functor cohomology: you have to have an acyclic covering–a covering on which the derived functor cohomology is trivial. But a priori, how can we tell that an open set is acyclic? What if we only know Cech cohomology? The point of today’s post is to use the heavy machinery of the Cech-to-derived functor spectral sequence to get such a purely Cech-theoretical criterion.

Cartan’s theorem gives a sufficient criterion for this to be the case. The result is:

Theorem 42 Let ${X}$ be a space, ${\mathcal{F}}$ a sheaf on ${X}$. Suppose there is a basis ${\mathfrak{A} }$ of open sets on ${X}$, closed under finite intersections, satisfying the following condition. If ${\mathfrak{B} \subset \mathfrak{A}}$ is a finite open covering of ${U \in \mathfrak{A}}$, then the Cech cohomology in positive dimension vanishes,$\displaystyle H^k(\mathfrak{B}, \mathcal{F})=0.$

Then the natural map:

$\displaystyle H^k(\mathfrak{A}, \mathcal{F}) \rightarrow H^k(X, \mathcal{F})$

is an isomorphism, for any ${k \in \mathbb{Z}_{\geq 0}.}$

I confess to having stated the result earlier incorrectly, when I claimed that the conclusion was ${H^k(X, \mathcal{F})=0}$ for ${k \geq 1}$.

But in any case, this will finally(!) complete the proof of the vanishing of the higher quasi-coherent cohomology of an affine. For then we just take ${\mathfrak{A}}$ to be the collection of basic open affines. We have shown that the Cech cohomology with respect to this family covers vanishes (on the whole space and on any basic open set, which is also affine!).

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

$\displaystyle \mathfrak{C}' \stackrel{F}{\rightarrow} \mathfrak{C} \stackrel{G}{\rightarrow} \mathbf{Ab}.$

Up until now, I have been talking primarily about the commutative algebra purely. I think I now want to go in a more algebro-geometric direction, partially because I find it easier to understand that way. Today, I will explain how the Koszul complex lets you compute certain types of Cech cohomology.

0.8. The Koszul complex and Cech cohomology

What we now want to show is that on a reasonable scheme, Cech cohomology of a quasi-coherent sheaf is really a type of Koszul cohomology. Namely, let’s start with a scheme ${X}$, which I will take to be quasi-compact and quasi-separated. (If you are what Ravi Vakil calls a noetherian person, then you can ignore the previous remark.)

Let ${\mathcal{F}}$ be a quasi-coherent sheaf on ${X}$. Let ${f_1, \dots, f_r \in \Gamma(X, \mathcal{O}_X)}$ be global regular functions on ${X}$. Then we can define the sets ${X_{f_i}}$ where the functions ${f_i}$ “don’t vanish” (more precisely, are units in the local ring). One of the basic results one proves is that taking sections over these basic open sets corresponds to localization:

Proposition 29 ${\Gamma(X_f, \mathcal{F}) = \Gamma(X, \mathcal{F})_f}$ if ${\mathcal{F}}$ is quasi-coherent.

Proof: This is a general fact about quasi-coherent sheaves, and one way to see it is to use the fact that if ${A = \Gamma(X, \mathcal{O}_X)}$ is the ring of global functions, there is a morphism ${g: X \rightarrow \mathrm{Spec} A}$. This is a quasi-separated, quasi-compact morphism by hypothesis. Thus the direct image ${g_*(\mathcal{F})}$ is quasi-coherent. In particular, this means that

$\displaystyle \Gamma(\mathrm{Spec} A, g_*(\mathcal{F}))_f = \Gamma(D(f), g_*(\mathcal{F}))$

where ${D(f) \subset \mathrm{Spec} A}$ is the basic open set. When one translates this back via the definition of ${f_*}$, one gets the proposition.

We now continue with the original question. So let ${M = \Gamma(X, \mathcal{F})}$ be the global sections of the sheaf ${\mathcal{F}}$. We have seen that ${M_{f_i}}$ is ${\Gamma(X_{f_i}, \mathcal{F})}$ for each ${i}$. Similarly, ${M_{f_{i_1} \dots f_{i_k}}}$ is ${\Gamma(X_{f_{i_1}} \cap \dots \cap X_{f_{i_k}}, \mathcal{F})}$ for any ${k}$-tuple of the ${f_i}$. To avoid triple subscripts, let us write ${U_{i_k}}$ instead of ${X_{f_{i_k}}}$. This is precisely what we need to consider the Cech cohomology with respect to the open sets ${\mathfrak{A} = \left\{X_{f_i}\right\}}$. (more…)

To continue, I am now going to have to use the language of sheaves. For it, and for all details I will omit here, I refer the reader to Charles Siegel’s post at Rigorous Trivialties and Hartshorne’s Algebraic Geometry. When I talk about sheaf cohomology, it will always be the derived functor cohomology. I will briefly review some of these ideas.

Sheaf cohomology

The basic properties of this are as follows.

First, if ${X}$ is a topological space and ${i \in \mathbb{Z}_{\geq 0}}$, then ${H^i(X, \cdot)}$ is a covariant additive functor from sheaves on ${X}$ to the category of abelian groups. We have

$\displaystyle H^0(X,\mathcal{F}) = \Gamma(X,\mathcal{F}),$

that is to say, the global sections. Also, if

$\displaystyle 0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0$

is a short exact sequence of sheaves, there is a long exact sequence

$\displaystyle H^i(X,\mathcal{F}) \rightarrow H^i(X, \mathcal{G})\rightarrow H^i(X, \mathcal{H}) \rightarrow H^{i+1}(X,\mathcal{F}) \rightarrow \dots .$

Finally, sheaf cohomology (except at 0) vanishes on injectives in the category of sheaves.

In other words, sheaf cohomology consists of the derived functors of the (left-exact) global section functor. (more…)