If

is a short exact sequence of sheaves on then for all open,

is exact, i.e., sections is a left exact functor (so we can derive it). It is not exact in general (refer to Example 1Example 1).

Because then by the construction of the kernel.

This follows from .

Pick . But is injective. So we have

and because is already a sheaf. So by injectivity of .

Therefore the global sections functor is left exact. Because has enough injectives, right derived functors exist.

Let and

where is the sheaf of analytic functions on , and is the sheaf of nowhere vanishing analytic functions. The sequence is exact because we can always take a local logarithm of a non-vanishing function, but such functions may not have logarithms globally. For example, on .

If where is open connected and -connected, then there exists such that .

We define

Let be a constant sheaf. If is “nice” (paracompact and locally contractible is OK), then

There is also a version for and de Rham cohomology. There are issues between connectedness vs path-connectedness: sheaf cohomology is built on recognising connected components, while singular cohomology is built out of simplices, which only detect path-components.

We can apply to the Example 1Example 1 to obtain a long exact sequence

If is simply connected, then is surjective.

Therefore . Also, this is another proof that is left exact.

Let be a fibre bundle with fibre and assume that is simply connected.

Now consider the composition

Then . Let be an abelian group. What is ? Locally in the fibre bundle, we just have

Assuming that all spaces are “nice”, then we should have

being simply connected means that locally constant sheaves are constant (we can’t go around in a loop and get something different). Therefore

This composition of left exact functors has a Grothendieck spectral sequence (Leray spectral sequence)

If , we end up with the Serre spectral sequence

One can compute derived functors using acyclic resolutions. Let be left exact and

is an -acyclic resolution, i.e., for all . Then we consider

The cohomology of this complex is .

Breaking the acyclic resolution up into short exact sequences, we obtain

The long exact sequence for gives

and also for each ,

where all the middle terms vanish because are -acyclic. Hence, we get isomorphisms

Looking more closely, we have a short exact sequence

So we have a long exact sequence

Therefore . Now because , we therefore have since is left exact, and one can check that

Exercise: .

Usually, acyclic resolutions are easier to compute than injective ones.

Example from vector calculus:

is an acyclic resolution of the constant sheaf .

If then the restriction map is surjective.

Usually sheaves of functions satisfy this (unless these functions are too constrained).

For nice , the sheafification of the presheaf of cochains is flasque: , where consists of -linear combinations of .