How does cohomology relate to cohomology?

Part 2

9th of August, 2026

1. Sheaves
2. The sheaf of holomorphic functions

Edit: Having now completed the full draft of this post, it turns out that I’m actually just going to talk about sheaves and holomorphic functions for a really long time. Stay tuned for the next part in this series, where I actually will get back to talking about cohomology!

In the last post, we saw “the” definition of cohomology, namely that of internal cohomology of a cochain complex \mathscr{H}^n(C^\bullet). From this definition, we then described how to build singular cohomology of a topological space \mathrm{H}^n(X;A). Maybe it will be useful to start recording these definitions in a table that we can add to as we go along. Let’s see what we have so far:

Ok, nice! Let’s add some more rows to this table, and let’s bring some sheaves into the story.

Sheaves

Just as I didn’t really want to end up writing a blog post about why we care so much about cohomology, I don’t really want to write one right now about sheaves. So, for fun, I’m going to start by giving “the other” definition of sheaf, for a change of pace. Then after a few paragraphs of digression I’ll give the usual, slightly less whimsical, definition so that we can get back to talking about what really matters: which type of “H” we’re going to use for our next cohomology definition.

What’s a sheaf? Well for the purposes of today’s post it’s just one space nicely indexed by another space: a sheaf Y on a base space X is a continuous map \pi\colon Y\to X that we impose a condition on. What condition, I hear you ask? Well, we’re not going to ask that \pi be a homeomorphism (i.e. a topological isomorphism) because then this wouldn’t be very interesting at all.¹ But we can ask that \pi be a little bit of a homeomorphism. For example, maybe it becomes a homeomorphism when we restrict it to some subset Y'\subset Y. That could be interesting, right? We wouldn’t be saying that Y\cong X, but merely that Y'\cong X' for some subsets Y'\subset Y and X'\subset X. Oh, how about this: there are actually lots of subsets of Y on which \pi restricts to a homeomorphism. Hmm, how many? That’s a good question. Let’s just go the whole way and say that there’s one (not necessarily unique) for each point in Y. And since we’re dealing with topological spaces, we should probably make sure that all of these subsets are actually open subsets.

Great! We’ve rediscovered the definition of a local homeomorphism: for all y\in Y there exists an open neighbourhood V_y\subseteq Y such that U_y\coloneqq\pi(V_y)\subseteq X is open and \pi|V_y is a homeomorphism \pi|V_y\colon V_y\cong U_y. We are then simply defining a sheaf on X as a local homeomorphism into X. Unfortunately, while this definition is very good for some purposes (e.g. defining pullback sheaves, or sheafification of a presheaf, or carrying out representability arguments) it is less convenient when we want to start throwing some algebra into the picture. So let’s switch back to the other definition of a sheaf.

We know that a local homeomorphism \pi\colon Y\to X lets us view Y as being made up of open subsets of X, at least locally. So since \pi isn’t a “global” homeomorphism, it doesn’t really make sense for us to try to find a “global” inverse to \pi, i.e. a map \sigma\colon X\to Y such that \pi\circ\sigma=\mathrm{id}_X and \sigma\circ\pi=\mathrm{id}_Y. But we could do exactly the same trick, and try to find “local” inverses. However, we will run into a problem here: there is a real asymmetry in the definition of local homeomorphism, and a rather nasty subtlety to some natural terminology. In fact, this is a problem that we see all over the place once we start looking at formalisms of so-called weak equivalences, namely that there is a very big difference between two objects “being weakly equivalent” and there actually existing a weak equivalence between them. More concretely, it seems (and is!) natural to say that Y is locally homeomorphic to X if every point y\in Y has an open neighbourhood V_y such that there exists some homeomorphism \pi_{V_y}\colon V_y\cong U_y\coloneqq\operatorname{Im}(\pi_{V_y})\subseteq X. This looks very similar to our definition of local homeomorphism but with one key difference: we did not specify some “global” map \pi\colon Y\to X at the start. So yes, if we have some local homeomorphism \pi\colon Y\to X then it is indeed true that Y is locally homeomorphic to X, but the converse does not necessarily hold, because we don’t know that we will be able to “glue together” all the \pi_{V_y} to get a map \pi\colon Y\to X. For example, if there exist some V_y and V_{y'} that overlap (V_{yy'}\coloneqq V_y\cap V_{y'}\neq\varnothing) but \pi_{V_y} and \pi_{V_{y'}} are not equal on this overlap (\pi_{V_y}|V_{yy'}\neq\pi_{V_{y'}}|V_{y''}) then how should we define our tentative map \pi on this overlap? Should it be equal to \pi_{V_y} or \pi_{V_{y'}}? Even if we make a choice, we now need to prove that we won’t run into any problems anywhere else with any other intersections, and then we need to worry about intersections of three open subsets, and then of four, and… and so on. The classical example of such a case is any sufficiently non-trivial manifold, say the sphere S^2\subset\mathbb{R}^3. By definition of being a manifold², it is indeed locally homeomorphic to \mathbb{R}^2, but there cannot³ exist any local homeomorphism S^2\to\mathbb{R}^2. As a bonus annoying quirk, note that there is a local homeomorphism in the other direction: \mathbb{R}^2\to S^2, given by the inverse⁴ to stereographic projection.

This asymmetry means (although I’m not suggesting that this is obvious) that we shouldn’t expect to be able to find even a local inverse to a local homeomorphism, but instead only a one-sided local inverse. That is, given a local homeomorphism \pi\colon Y\to X, we might have some success in finding maps X\supseteq U\xrightarrow{\sigma_U} V\subseteq Y such that \pi\circ\sigma_U=\mathrm{id}_U. So let’s turn this into a definition! Given a sheaf \pi\colon Y\to X and an open subset U\subset X, we define the set of sections of Y\xrightarrow{\pi}X over U as \Gamma(U,Y\xrightarrow{\pi}X) \coloneqq \{\sigma\colon U\to Y \mid \pi\circ\sigma = \mathrm{id}_U\}. But if we leave out any specific choice of open subset U we obtain a (contravariant) functor from the category (really, the poset; morphisms are given by inclusion of one open subset into another, and this doesn’t involve any choice or extra data) \mathsf{Op}(X) of open subsets of X to the category of sets: \begin{aligned} \Gamma\colon \mathsf{Op}(X)^\mathrm{op} &\longrightarrow \mathsf{Set} \\ U &\longmapsto \Gamma(U,Y\xrightarrow{\pi}X) \end{aligned} This functor is particularly nice, since it satisfies exactly the kinds of properties that we were worried about before when asking whether or not some \pi_{V_y} and \pi_{V_{y'}} would be able to “glue” on the overlap V_{yy'}. We could phrase this property in terms of “unique gluing of compatible local sections”, or in terms of “descent”, but I’m actually going to opt for neither of those and just keep things a bit vague. That is, a sheaf (from this perspective) is a presheaf (i.e. a functor \mathsf{Op}(X)^\mathrm{op}\to\mathsf{Set}) that comes from this construction of starting with a local homeomorphism.

If this were a real introduction to sheaves, then I would talk about structure sheaves and, in particular, the sheaf of holomorphic functions, which is sort of “the original” sheaf. However, this is not a real introduction to sheaves! We’ll be content for now to just point out one generalisation: what if we replace the category \mathsf{Set} by something with algebraic structure, e.g. the category of abelian groups? If we do this then we get sheaves of abelian groups, and these are the guys that we really care about in the story of sheaf cohomology. To give just one concrete example, I will actually talk a tiny bit about the sheaf of holomorphic functions. If you don’t really know about holomorphic functions, then don’t worry, because I’ll just say some things that I promise are true and you can just believe me and get on with your life.

The sheaf of holomorphic functions

To pick one of the myriad of (eventually equivalent) definitions, a holomorphic function f\colon\mathbb{C}\to\mathbb{C} is a function that is everywhere complex-differentiable. That is, for all z_0\in\mathbb{C}, the limit f'(z_0) \coloneqq \lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0} exists, i.e. is a well-defined complex number, i.e. gives a well-defined function f'\colon\mathbb{C}\to\mathbb{C}. The thing that makes holomorphic functions so wonderful (and also so tough to work with) is the sneaky little detail about what we mean when we say “the limit as z approaches z_0”. In real analysis, working in \mathbb{R}, there are basically just two ways that one number x can approach another number x_0: either x gets smaller and smaller until it is really close to x, or it gets larger and larger until it is really close to x. That is, \mathbb{R} just looks like a line. But \mathbb{C} is a whole plane, and this means that z can approach z_0 in infinitely many interesting ways. For example, it could approach it along any straight line coming out of z_0, or it could “spiral inwards”, or do something even more fancy. The fact that there are so many possible ways that z could approach z_0 means that this limit existing is a really strict requirement. And, as is usual in mathematics, the more strictly defined an object is, the more well behaved it is. This leads to a bunch of other (again, equivalent) definitions of holomorphic functions, and we’ll focus on one in particular: a function f\colon\mathbb{C}\to\mathbb{C} is holomorphic if and only if it is analytic, i.e. if an only if, for all z_0\in\mathbb{C}, we can find a (nice) open neighbourhood U_{z_0}\subseteq\mathbb{C} on which f can be expressed as a power series centred at z_0: f|U_{z_0} = \sum_{i=0}^\infty a_i(z-z_0)^i = a_0 + a_1(z-z_0) + a_2(z-z_0)^2 + \ldots.

Lots of nice functions are holomorphic. For example, any polynomial is holomorphic: in short, a polynomial is just a finite power series, so it’s definitely analytic. All the trigonometric functions that we know and love (when suitably extended from being defined only on the real numbers to being defined on all the complex numbers) are holomorphic, and this is because the complex exponential function z\mapsto e^{\lambda z} is holomorphic (for any fixed \lambda\in\mathbb{C}). Holomorphic functions are also closed under all the sensible operations: point-wise addition, subtraction, product, and multiplication by a scalar (and thus form a ring, and a \mathbb{C}-vector space; they form a \mathbb{C}-algebra), and even division, as long as the denominator is never zero anywhere.⁵ They’re also closed under composition.

However, holomorphic functions are very “rigid”. Here are two fundamental theorems that demonstrate what we mean when we say that:

So Picard’s little theorem tells us that non-constant holomorphic functions cannot have their image squished down into some small open subset; holomorphic functions really have to have complete free roam of the complex plane. In fact, it generalises another fundamental theorem, namely Liouville’s theorem, which says that any holomorphic function that is bounded must actually in fact be constant. Liouville’s theorem, although weaker, can actually be used to prove the fundamental theorem of algebra, as well as another “rigidity” property of holomorphic functions: if f and g are holomorphic functions, and g dominates f (i.e. |f(z)|\leqslant|g(z)| for all z\in\mathbb{C}), then actually g and f are basically just the same function (i.e. there exists some \lambda\in\mathbb{C} such that f=\lambda g).

The identity theorem tells us, in particular, that if f is constant on any connected open subset of \mathbb{C}, no matter how small, then it is globally constant. This is really strong: the global behaviour of a holomorphic function can be entirely determined by its behaviour on the smallest possible open disc you could imagine! Indeed, it turns out that a holomorphic function (defined on any non-empty connected open set) is always entirely determined by its restriction to any non-empty open subset. This is very very not true for real analytic functions, or smooth functions, or anything familiar back in the world of real analysis!

These facts have a lot of consequences for the geometry of holomorphic functions. We can define holomorphic functions of many variables, i.e. f\colon\mathbb{C}^n\to\mathbb{C}, in a way that turns out to be surprisingly simple: Hartog’s theorem (or Osgood’s lemma, if we are a bit more careful) tells us that we just need to ask for f to be holomorphic in each variable. This is surprising because the same is not true in real analysis! The famous⁶ counter-example is f(x,y) \coloneqq \begin{cases} \frac{xy}{x^2+y^2} &\text{if }(x,y)\neq(0,0) \\ 0 &\text{if }(x,y)=(0,0) \end{cases} which is differentiable in each variable, but it is not even continuous (at the origin) in both variables. Once we know how to talk about multivariable holomorphic functions, we can define complex manifolds as topological spaces that are locally biholomorphic to (i.e. there exists a holomorphic map with holomorphic inverse) open⁷ subsets of \mathbb{C}^n. And now, finally, we can get back to the sheaf of holomorphic functions.

If we have any space X whatsoever, then it’s usually interesting to study the functions on that space⁸, by which we mean maps X\to A, where A is some suitably relevant space, and where the maps preserve whatever interesting structure is common between X and A. For example, in the setting of complex geometry, we are interested in holomorphic functions from a complex manifold into the complex plane: f\colon X\to\mathbb{C}. But now these rigidity theorems come into play: if X is a compact complex manifold (e.g. complex projective space, or the vanishing of some (homogeneous) polynomials inside complex projective space) then every holomorphic function X\to\mathbb{C} is actually just a constant function. This is bad news! Looking at functions on a space no longer helps us recover that space up to isomorphism, and that will make a lot of category theorists very sad. This will also make a lot of real-analysis fans sad, because there are lots and lots of smooth functions from a smooth manifold into \mathbb{R}.

There are two (not mutually exclusive) ways of fixing this.

One way is to say “well maybe we should just ignore the compact spaces”. This is actually a surprisingly good idea, in some senses. It is true that this means we will be leaving the world of projective geometry, and thus throwing away a gargantuan bunch of useful tools that Chow’s theorem (or, more generally, GAGA) would have let us borrow from algebraic geometry. It’s also true that we’re throwing away a hypothesis that makes a lot of things a lot easier (finite things are so much simpler than infinite things!). But the reward is that we’ll start seeing a lot more Stein manifolds around, and these are my best friend. I actually have a draft blog post about Stein manifolds, non-algebraic complex geometry, and model categories, so I’m going to just leave this sentence as an unexplained cliffhanger for now.

The other way is to go back to sheaf theory, and so this is the one that we’ll be exploring here. It turns out, by the above theorems about holomorphic functions, and a little bit more complex analysis, that if we have some complex manifold X (compact or not, we now don’t care) then around any point x\in X we can always find a small enough open neighbourhood U_x\subseteq X such that the set of holomorphic functions on U_x is actually the same as the set of holomorphic functions on \mathbb{C}^n. That’s a lot of holomorphic functions! If anything, that’s maybe too many holomorphic functions… When we have some really small U_x and another really small U_{x'}, what are we going to do on U_{xx'}\coloneqq U_x\cap U_{x'}? There are two ways we can think of \{f\colon U_{xx'}\to\mathbb{C}\} as being equal to \{f\colon\mathbb{C}^n\to\mathbb{C}\}, and what if they’re not the same? But hopefully some alarm bells are ringing in your head now: we’ve already had this exact same conversation! Let’s build a sheaf!

Given a complex manifold X we define its sheaf of holomorphic functions (usually denoted by \mathcal{O}_X) to be given by \begin{aligned} \mathcal{O}_X\colon \mathsf{Op}(X) &\longrightarrow \mathsf{Ab} \\ U &\longmapsto \{f\colon U\to\mathbb{C} \mid f\text{ is holomorphic}\} \end{aligned} Note that we could have chosen to land in the category of \mathbb{C}-algebras, but we’ll just look at the underlying abelian group for now. But why is this thing a sheaf? That’s an argument we’re not going to get into today. I’m just going to say that it is, and you can either just believe me or (much better) think about it and do some internet/book searching. But the argument that needs to be given is, in essence, that if we have open subsets U,V\subseteq X with a non-empty overlap U\cap V\neq\varnothing, and holomorphic functions f_U\colon U\to\mathbb{C} and f_V\colon V\to\mathbb{C} such that f_U=f_V, then there actually exists a unique holomorphic function f_{U\cup V}\colon U\cup V on the union such that f_U=f_{U\cup V}|U and f_V=f_{U\cup V}|V. That is, if we have a collection of holomorphic functions on little open subsets, and whenever they could possibly be simultaneously defined they actually all agree, then we never really had a collection of little holomorphic functions but instead just secretly had one big holomorphic function the whole time, that recovers the little ones by restriction.

This sheaf is amazing. In fact, it’s so amazing that there’s a whole theorem just saying how amazing this sheaf is, and it’s such an incredible theorem that I would argue that it deserves the name of “fundamental theorem of complex geometry”. In fact, because this is our blog post and we can do what we like, let’s just make that true now.

If we wanted to use some more standard terminology here, we would substitute the word “amazing” for the word “coherent”, and our theorem then becomes the following:

The story of coherent sheaves, however, really will have to wait for another day.