Sheaf (Čech) Cohomology: A glimpse

This is a blogpost on Sheaf Cohomology. We shall be following this article.

If the reader wants to read up on what a sheaf is, he/she can read the very readable wikipedia article on it.

From the word cohomology, we can guess that we shall be talking about a complex with abelian groups and boundary operators. Let us specify what these abelian groups are.

Given an open cover \mathcal{U}=(U_i)_{i\in I} and a sheaf \mathcal{F}, we define the 0^{th} cochain group C^0(\mathcal{U}, \mathcal{F})=\prod_{i\in I}\mathcal{F}(U_i). Note that we are not assuming that the sections over the individual U_i‘s agree on the intersections. This is simply a tuple in which each coordinate is a section. We are interested in finding out whether we can glue these sections together to get a global section. This is only possible if the sections agree on the intersections of the open sets.

We now define C^1(\mathcal{U}, \mathcal{F})=\prod_{i,j\in I}\mathcal{F}(U_i\cap U_j). Here we are considering the tuple of sections defined on the intersections of two sets. Note that these intersections may not cover the whole of the topological space. Hence, we are no longer interested in gluing sections together to see whether they form a global section.

Similarly, we define C^2(\mathcal{U}, \mathcal{F})=\prod_{i,j,k\in I}\mathcal{F}(U_i\cap U_j\cap U_k).

Now, we come to the boundary maps. \delta: C^0(\mathcal{U}, \mathcal{F})\to C^1(\mathcal{U}, \mathcal{F}) is defined in the following way: \delta(f_i)=(g_{i,j}), where (g_{i,j})= f_{j|U_i\cap U_j}-f_{i|U_i\cap U_j}. What we’re doing is that we’re taking a tuple of sections, and mapping it to another tuple; the second tuple is generated by choosing two indices k,l, determining the sections defined over U_k and U_l, and then calculating f_{l|U_k\cap U_l}-f_{k|U_k\cap U_l}. In the image tuple, f_{l|U_k\cap U_l}-f_{k|U_k\cap U_l} would be written at the k,l coordinate.

Now we define the second boundary map. \delta: C^1(\mathcal{U}, \mathcal{F})\to C^2(\mathcal{U}, \mathcal{F}) is defined in the following way: \delta(f_{ij})=(g_{i,j,k}), where (g_{i,j,k})= f_{i,j|U_i\cap U_j\cap U_k}-f_{k,i|U_i\cap U_j\cap U_k}+f_{j,k|U_i\cap U_j\cap U_k}. What does this seemingly arbitrary definition signify? The first thing to notice is that if f_{i,j} is an image of an element in C^0(\mathcal{U}, \mathcal{F}), then \delta(f_{i,j})=0. Hence, at the very least, this definition of a boundary map gives us a complex on our hands. Maybe that is all that it signifies. We’re looking for definitions of C^i(\mathcal{U},\mathcal{F}) which keep us giving sections over smaller and smaller open sets, and definitions of \delta over these C^i(\mathcal{U},\mathcal{F}) which keep on mapping images from C^{i-1}(\mathcal{U},\mathcal{F}) to 0.

Predictably, H^i(\mathcal{U},\mathcal{F})=Z(\mathcal{U},\mathcal{F})/B^i(\mathcal{U},\mathcal{F}), where Z(\mathcal{U},\mathcal{F}) is the kernel of \delta acting on C^i(\mathcal{U},\mathcal{F}) and B^i(\mathcal{U},\mathcal{F}) is the image of \delta acting on C^{i-1}(\mathcal{U},\mathcal{F}). Sheaf cohomology, measures the extent to which tuples of sections over an open cover fail to be global sections. The longer the non-zero tail of the cohomology complex, the farther the sections of this sheaf lie from gluing together amicably. In other words, the length of the non-zero tail measures how “complex” the topological space and the sheaf on it are. However, there is still hope. By a theorem of Grothendieck, we know that the length of the complex is bounded by the dimension of the (noetherian) topological space.