Cohomology as a measure of local to global failure

Posted: Mon 25 December 2017
Filed under mathematics
Tags: cohomology sheaves topology

Motivation for cohomology

In most introductory algebraic topology courses, cohomology is rather poorly motivated. It's most commonly seen form in an algebraic topology course is singular cohomology, which arises as a the homology of the dual of the singular chain complex, but that doesn't really tell you why it's of any more interest than singular homology, aside from the fact that you get an additional cup product which you did not have before. However, this approach obscures the geometric meaning of cohomology.

A much more intuitive introduction to cohomology turns out to be De Rham cohomology, often encountered in introductory differential geometry courses. Loosely speaking, De Rham cohomology measures in how many different ways can a closed form fail to be exact.

Another cohomology theory we'll look at is sheaf cohomology. Loosely, the cohomology of a sheaf measures how a certain functor $\Gamma$, which we'll define later fails to be an exact functor.

In the case of De Rham cohomology, we'll interpret the form being closed as a local property, and it being exact a global property, and the the case of sheaf cohomology, the exactness of the following sequence (the exact details of which we'll see in a later section) as a local property.

$$0 \rightarrow \mathcal{S}_1 \rightarrow \mathcal{S}_2\rightarrow \mathcal{S}_3 \rightarrow 0$$

The exactness of the sequence we get by applying $\Gamma$ to the above sequence turns out to be a global property.

In both the cases, the cohomology measures how the local property fails to translate to the global one.

De Rham Cohomology

The De Rham cohomology of a manifold $M$ (of dimension $m$) is the homology of the following of the following cochain sequence.

$$0 \xrightarrow{d} \Lambda^0(M) \xrightarrow{d} \Lambda^1(M) \xrightarrow{d} \cdots \xrightarrow{d} \Lambda^m(M) \xrightarrow{d} 0$$

Here, $\Lambda^k(M)$ is the space of $k$-forms on $M$, and $d$ is the exterior derivative operator. For a $k$-form $\omega$ to be closed, $d\omega$ must be $0$. This is a local property, in the sense that $d\omega$ evaluated at any point $p \in M$ depends only on the value of $\omega$ on any small neighbourhood of $p$. In fact, one can say a little more, and claim that $d\omega(p)$ depends only on the germ of $\omega$ at $p$. If we pick a Euclidean neighbourhood $U$ of $p$ which is homeomorphic to the open unit ball, the Poincaré lemma tells us that there is some $(k+1)$-form $\eta$ defined on $U$ such that $\omega = d\eta$. In other words, the closed form $\omega$ is locally exact.

The De Rham cohomology class of $\omega$ measures how badly does the property of local exactness fail to translate to global exactness. We can write $\omega$ as $\gamma + d\zeta$, where $\gamma$ is the canonical representative of the cohomology class of $\omega$ (more on this in later posts), and $\zeta$ is a $(k-1)$-form. If we stretch the analogy a bit, we can say $\omega$ misses being globally exact by $\gamma$ amount. This is the first example of how cohomology measures how badly a local property fails to be global.

Sheaf Cohomology

Before we see what sort of local to global failure sheaf cohomology measures, we'll quickly define sheaves and sheaf cohomology, and look at one example.

Quick introduction to sheaves

Given a manifold $M$ (whatever we discuss will hold in for Hausdorff spaces, and with a little more work, can be made to work even for a larger class of spaces like spectra of rings), a sheaf $\mathcal{S}$ of $K$-modules ($K$ is always assumed to be a commutative ring with identity) over $M$ is a topological space $\mathcal{S}$ with a surjective map $\pi: \mathcal{S} \to M$, such that the following properties are satisfied.

$\pi$ is a local homeomorphism, i.e. for any point $s \in \mathcal{S}$, there's a neighbourhood of $s$ such that $\pi$ restricted to that neighbourhood is a homeomorphism.
$\pi^{-1}(x)$, which we'll denote by $\mathcal{O}_x$, is a $K$-module, for all $x \in M$. $\mathcal{O}_x$ is called the stalk of $\mathcal{S}$ at $x$.
The module operations on the stalk are continuous, i.e. if we look at the stalk with the subspace topology, the module operations of addition and scalar multiplication are continuous.

Sheaves in some sense a modules parametrized by the space $M$, like vector bundles, but vector bundles do not satisfy the first condition, unlike sheaves. The simplest example of a sheaf is the constant sheaf which is just $M \times V$, where $V$ is a $K$-module with the discrete topology.

Another important example is the sheaf of germs of $C^{\infty}$ functions on a manifold $M$. For each $x \in M$, a point in $\mathcal{O}_x$ is an equivalence class of functions, the equivalence relation being that $f \sim g$ if $f$ and $g$ agree on some neighbourhood of $x$. This sheaf deserves a post of its own, and I shall write about it in the future.

The last example, which will be key to our goal, is the skyscraper sheaf. We'll describe it by first describing the stalk at each point, and then putting an appropriate topology on it. Fix a point $x_0 \in M$. The stalk $\mathcal{O}_{x_0}$ at $x_0$ will be $K$ as a module over itself. The stalk at every other point is the zero module. As a set, our sheaf is the following.

$$\mathcal{S} = K \sqcup \bigsqcup_{x \neq x_0} \{0\}$$

The question is what topology do we put on this space. The line with two origins provides a hint. What we do is take $|K|$ copies of the space $M$, and if $x \neq x_0$, we identify all of those $x$'s, otherwise we do nothing. It's not too hard to check that this defines a sheaf over $M$ (the local homeomorphism property is the hardest to check, and relies on the fact that points are closed in Hausdorff spaces). In fact, if $M = \mathbb{R}$ and $K = \mathbb{Z}/2$, then then skyscraper sheaf at $0$ is the line with two origins. The reason this is called the skyscraper sheaf is because only the stalk at $x_0$ is tall, the stalks everywhere are flat, which makes it look like a tall structure in an otherwise flat featureless landscape.

We're really interested in is a variant of a skyscraper sheaf with two skyscrapers, i.e. the stalks at points $x_0$ and $x_1$ are $K$, and otherwise $0$. The topology on this sheaf can be defined analogously. We'll come back to this example once we've defined sheaf cohomology.

The category of sheaves of $K$-modules over a space

Just like in the case of a vector bundles over a manifold $M$, where the right kind of map between vector bundles is a smooth map that is a linear map on each fibre, the right kind of map between two sheaves $\mathcal{S}_1$ and $\mathcal{S}_2$ on a space $M$ is a continuous map $f$ such that it satisfies the following properties.

$\pi = \pi \circ f$
$f$ restricted to any any stalk $\mathcal{O}_x$ is a $K$-module homomorphism.

Fixing a space $M$, we get the category of sheaves of $K$-modules over $M$, whose objects are sheaves, and the morphisms are what we just defined, called sheaf homomorphisms. It follows from the fact that $\pi$ is a local homeomorphism that even sheaf homomorphisms are local homeomorphisms. This category turns out to be especially nice, sharing many characteristics with the category of abelian groups and more generally, the category of $K$-modules, such as maps possessing kernels and cokernels, and possessing a version of the First Isomorphism Theorem. This sort of category is called an abelian category, and this category is the appropriate category to do homological algebra in. Coming back to sheaves, the kernel of a sheaf homomorphism $f: \mathcal{S}_1 \to \mathcal{S}_2$ is the set of all points which map to the zero element in the stalk. With a little bit of work, we can show the image of $f$ is a sheaf in its own right, and subsheaf of $\mathcal{S}_2$, just like the kernel of $f$ is a subsheaf of $\mathcal{S}_1$ (the definition of a subsheaf is the most obvious one).

With all these definitions in hand, we can talk about exact sequences of sheaves. Consider a sequence of sheaves and sheaf homomorphisms of the following kind.

$$\cdots \xrightarrow{d_{i-2}} \mathcal{S}_{i-1} \xrightarrow{d_{i-1}} \mathcal{S}_{i} \xrightarrow{d_{i}} \mathcal{S}_{i+1} \xrightarrow{d_{i+1}} \cdots$$

This sequence is exact if $\mathrm{ker}(d_{i}) = \mathrm{im}(d_{i-1})$.

The next thing we look at is the functor $\Gamma$ from the category of sheaves of $K$-modules over $M$ to the category of $K$-modules. For each sheaf $\mathcal{S}$, the object $\Gamma(\mathcal{S})$ is the module of sections of $\mathcal{S}$. A section of a sheaf $\mathcal{S}$ is a map $s: M \to \mathcal{S}$ such that $\pi \circ s = \mathrm{id}$. Clearly, we can add two sections, and we can also multiply them by a scalar; we therefore have a $K$-module. The functor $\Gamma$ acts on morphisms by composing them with the section map, i.e. $\Gamma(f) = f \circ s$. The important question to ask here is whether the functor $\Gamma$ is exact, i.e. does it short exact sequences to short exact sequences. The answer is no. Consider the following short exact sequence.

$$0 \rightarrow \mathcal{S}_1 \xrightarrow{\alpha} \mathcal{S}_2\xrightarrow{\beta} \mathcal{S}_3 \rightarrow 0 \tag{1}$$

If we apply the functor $\Gamma$ to the sequence, we get something that is not completely exact.

$$0 \rightarrow \Gamma(\mathcal{S}_1) \xrightarrow{\Gamma(\alpha)} \Gamma(\mathcal{S}_2) \xrightarrow{\Gamma(\beta)} \Gamma(\mathcal{S}_3) \rightarrow 0 \tag{2}$$

This sequence is exact only exact at $\Gamma(\mathcal{S}_1)$ and $\Gamma(\mathcal{S}_2)$.

Suppose some $s \in \Gamma(\mathcal{S}_1)$ maps to $0$ in $\Gamma(\mathcal{S}_2)$. That tells us that $\Gamma(\alpha)(s) = 0$. But that by definition means that $\alpha \circ s = 0$. But $\alpha$ is injective, which means $s = 0$. This shows exactness at $\Gamma(\mathcal{S}_1)$.

Showing exactness at $\Gamma(\mathcal{S}_2)$ is a little more involved. Consider an element $s \in \Gamma(\mathcal{S}_2)$ which gets mapped to the zero section in $\Gamma(\mathcal{S}_3)$. That means for all $m \in M$, $\beta(s(m)) = 0$. By exactness at $\mathcal{S}_2$, we can find for each $m$, an element $s'(m)$ of $\mathcal{S}_1$ such that $\alpha(s'(m)) = s(m)$. Furthermore, because the original short exact sequence is exact at $\mathcal{S}_1$, the element $s'(m)$ is uniquely defined (this is where the argument fails to work for $\Gamma(\mathcal{S}_3)$). All we need to show now is that the map $m \mapsto s'(m)$ is a continuous map. This is where we use the fact that sheaf homomorphisms are local homeomorphisms. For any $m$, pick a small enough neighbourhood $U$ around $s'(m)$ such that $\alpha$ is a local homeomorphism on $U$. Then $s'^{-1}(U)$ is given by $s^{-1}(\alpha(U))$, which is open since $s$ is a continuous section.

Notice that the exactness of sequence $(1)$ is a purely local property; it suffices to check whether the sequence on each stalk is exact. On the other hand, showing exactness at $\Gamma(\mathcal{S}_3)$ would be a global property. This is because given any section $s \in \Gamma(\mathcal{S}_3)$, the best we can do is construct sections $s_U$ on open subsets $U$ of $M$. It might so happen that these sections defined on different subsets of $M$ cannot be patched together consistently to get a continuous section. The cohomology of the sheaf will measure how badly the functor $\Gamma$ fails to be exact; to be more precise, the cohomology will tell us how extend sequence $(2)$ to get an exact sequence. We'll leave the precise details of this for a later post, and satisfy ourselves with an example of when exactness fails to happen at $\Gamma(\mathcal{S}_3)$.

To show this, we will exhibit a surjective sheaf homomorphism $f$ such that $\Gamma(f)$ is not a surjective module map. Consider a connected space $M$, and let $\mathcal{S}_1$ be the constant sheaf on $M$. Recall that this means $\mathcal{S}_1$ is $M \times K$, with the discrete topology on $K$. Let $\mathcal{S}_2$ be the skyscraper sheaf on $M$ with two skyscrapers, which means the stalk is $K$ at points $x_0$ and $x_1$ and zero otherwise. On the stalk at point which is not $x_0$ or $x_1$, the homomorphism is obviously the zero homomorphism. On the stalk at $x_0$ and $x_1$, we let the homomorphism be the identity homomorphism. It's clear that this sheaf homomorphism, call it $f$ is surjective. But observe that $\Gamma(\mathcal{S}_1) = K$. That's because we picked $M$ to be a connected manifold, which means the section must the constant section. On the other hand, $\Gamma(\mathcal{S}_2) = K \oplus K$, since the section can take any value independently at $x_0$ and $x_1$. Which means $\Gamma(f)$ is a map from $K$ to $K \oplus K$, which cannot be surjective in general.

This tells us that exactness at $\Gamma(\mathcal{S}_3)$ is a global property, and the cohomology measures (in a loose sense) how the local property of exactness of $(1)$ fails to translate to exactness of $(2)$.

ADDENDUM: I will add links to similar expositions whenever I find them.

Čech cohomology and the Mittag-Leffler problem: The Čech cohomology determines whether meromorphic functions defined on small open sets can be patched together to get a globally defined meromorphic function satisfying certain properties. (Link to article)

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