Koszul cohomology and Cech cohomology

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 ${\mathfrak{A} = \left\{X_{f_i}\right\}}$ .

A priori, note that the ${X_{f_i}}$ won’t cover ${X}$ , so we shouldn’t immediately expect that the Cech cohomology will resemble the derived functor cohomology. Well, so what is it? Recall that the ${k-1}$ -th part of the Cech complex is the set of ${k}$ -cochains, which associate to every ordered tuple ${i_1, \dots, i_k}$ an element in ${\Gamma(U_{i_1} \cap \dots U_{i_k}, \mathcal{F}) = M_{f_{i_1} \dots f_{i_k}}}$ . This cochain is required to be alternating, i.e. a swap in ${i_1, \dots, i_k}$ should flip a sign. In other words, the ${k-1}$ -th part is the subset of the product

$\displaystyle \prod_{i_1 \dots i_k} M_{f_{i_1}}$

consisting of alternating tuples. Let ${[1,r]}$ be the set of integers between ${1}$ and ${r}$ , and let ${\phi}$ be a ${k-1}$ -cochain in the Cech complex, so ${\phi}$ is an alternating function out of ${[1, r]^k}$ . The value on ${(i_1, \dots, i_k)}$ lies in the localization ${M_{f_{i_1} \dots f_{i_k}}}$ . The boundary of ${\phi}$ on a ${k+1}$ -tuple ${(i_1, \dots, i_{k+1})}$ is defined as

$\displaystyle \partial \phi(i_1, \dots, i_{k+1}) = \sum (-1)^s \phi(i_1, \dots, \hat{i_s}, \dots, i_{k+1}).$

So we have a fairly explicit description of the Cech complex in our case. This doesn’t look very much like a Koszul complex. For one thing, there are all sorts of localizations floating around. We can deal with this by the following result.

Lemma 30 Let ${A}$ be a ring, and ${M}$ an ${A}$ -module. Let ${f \in A}$ . Then ${M_f}$ is the direct limit of the system $\displaystyle M \stackrel{f}{\rightarrow} M \stackrel{f}{\rightarrow} M \dots.$

Proof: I’m only going to sketch the proof. The idea is that some ${m}$ in the ${i}$ th copy of ${M}$ should map to ${m/f^i}$ in ${M_f}$ . This is clearly compatible with the maps between the directed system, and one can check that the induced map is injective and surjective.

One amusing corollary, incidentally, is that a cocontinuous (or even one commuting with filtered colimits) functor from ${A}$ -modules to ${A}$ -modules preserves localization. Indeed, the above argument shows that it preserves localization at an element, and localization at a multiplicative set is a filtered colimit of localizations at various elements.

So, let’s see what this entails for the Cech complex. Instead of thinking of cochains ${\phi: [1,r]^k \rightarrow \sqcup \left\{\mathrm{various localizations}\right\}}$ , we now think of alternating cochains taking values in ${M}$ . Rather, we should take a direct limit over ${m}$ .

Let’s state this again. For each ${m}$ , we have a map ${M \rightarrow M_{f_{i_1} \dots f_{i_k}}}$ sending ${m \rightarrow m/(f_{i_1} \dots f_{i_k})^m}$ . There is a system of maps ${M \rightarrow M \rightarrow M}$ where each is multiplication by ${f_{i_1} \dots f_{i_k}}$ . At the ${m}$ -th stage, the natural inclusion ${M_{f_{i_1} \dots f_{i_k}} \rightarrow M_{f_{i_1} \dots f_{i_k} f}}$ corresponds to multiplication by ${f^m}$ , for any ${f \in R}$ . The direct limit of all these modules ${M}$ and all these morphisms gives the localizations ${M_{f_{i_1} \dots f_{i_r}}}$ and the natural maps between the localizations.

For each ${m}$ , let us consider the ${R}$ -module of alternating maps ${\overline{\phi}: [1,r]^k \rightarrow M}$ , which maps into the ${k-1}$ -th Cech cochain complex by identifying ${\overline{\phi}}$ with the ${k-1}$ -Cech cochain ${\phi}$ defined via ${\phi(i_1, \dots, i_r) = \overline{\phi}(i_1, \dots, i_r)/ (f_{i_1}\dots f_{i_r})^m}$ . Clearly the ${k}$ th cochain module is the limit of these. Note, however, that the differential has to be described differently. Namely, using the above identifications, we must have

$\displaystyle \partial \overline{\phi}(i_1, \dots, i_{r+1}) = \sum (-1)^s f_{i_s}^m \overline{\phi}(i_1, \dots, \hat{i_s}, \dots, i_{r+1}).$

To put everything together:

Proposition 31 The Cech complex of ${\mathcal{F}}$ with respect to the open cover ${\mathfrak{A} = \left\{X_{f_i}\right\}}$ is isomorphic (up to a shift) to the direct limit of the complexes ${C_m}$ whose ${k}$ th term consists of alternating maps ${\overline{\phi}: [1, \dots, r]^k \rightarrow M}$ and such that the boundary is $\displaystyle \partial \overline{\phi}(i_1, \dots, i_{r+1} ) = \sum (-1)^s f_{i_s}^m \overline{\phi}(i_1, \dots, \hat{i_s}, \dots, i_{r+1}).$

The morphisms ${\psi_m: C_m \rightarrow C_{m+1}}$ between the complexes are given by $\displaystyle (\psi \overline{\phi})(i_1, \dots, i_r) = f_{i_1}\dots f_{i_r} \overline{\phi}(i_1, \dots, i_r).$

Proof: This requires a little checking, which is probably best done at least partially for oneself. We have shown that the Cech complex is indeed the direct limit of this whole system, so all that is left to check is that these maps ${C_m \rightarrow C_{m+1}}$ are indeed maps of complexes. We shall show in fact that these are maps of complexes. In other words, what we need to see is:

Proposition 32 ${C_m}$ as described is the (truncated) Koszul complex ${K^*( f_1^m, \dots, f_r^m,M)}$ shifted by one. The maps ${C_m \rightarrow C_{m+1}}$ are natural maps of complexes.

The point of the truncation is that the shifted Koszul complex will be nonzero in dimension $-1$ , and we want to remove that.

Proof: Notice that the ${k}$ th term of the Koszul complex ${K_*(f_1^m, \dots, f_r^m, R)}$ is the free ${R}$ -module on objects ${e_{i_1} \wedge \dots \wedge e_{i_k}}$ . So a map from ${K^k(f_1^m, \dots, f_r^m)}$ into ${M}$ is just the same thing as an alternating map ${\phi: [1, r]^k \rightarrow M}$ . Since the boundary on the Koszul complex is defined via

$\displaystyle \partial (e_1 \wedge \dots \wedge e_r) = \sum f_i^m e_1 \wedge \dots \hat{e_i} \wedge \dots e_r,$

it is easy to see that the boundary described above on ${C_m}$ is just the dualized Koszul boundary map.

Finally, we must describe the morphism ${C_m \rightarrow C_{m+1}}$ and show that it is a morphism of complexes, in terms of the Koszul complex. Let ${\mathbf{f}, \mathbf{g}}$ be ${r}$ -tuples of elements in ${R}$ . I claim that there is a morphism of differential graded algebras

$\displaystyle K_*(\mathbf{fg}) \rightarrow K_*(\mathbf{f})$

(which induces a morphism on Koszul complexes with coefficients in any module ${M}$ ). This morphism is given by the natural extension of the diagonal morphism ${R^r \stackrel{(g_1, \dots, g_r)}{\rightarrow} R^r}$ . One can see directly that this commutes with the boundary map. Dualizing this gives a map

$\displaystyle K^*(\mathbf{f}, M) \rightarrow K^*(\mathbf{fg}, M),$

which is just the map ${C_m \rightarrow C_{m+1}}$ . I realize that I’ve been a little loose with the proofs, since they are nothing but straightforward verification, but I think I have sketched all the main ideas.

Anyway, the main idea to take away from this is that Cech cohomology over the open cover ${\mathfrak{A}}$ is a direct limit of Koszul cohomologies ${H^*(\mathbf{f}^m, M)}$ , with the dimension shifted by one. In particular, if $k$ is at least 2, ${H^{k-1}(\mathfrak{A}, \mathcal{F}) = \varinjlim_m H^{k}(\mathbf{f}^m, M)}$ . This will allow us to prove results in algebraic geometry using properties of the Koszul complex. Next time, we shall do the cohomology of affine space, for instance.