Recently in between studying for my finals I’ve been learning about a whole load of different things in algebraic geometry, a main one being cohomology in its various forms (sheaf, Čech, de Rahm, … even a bit of étale cohomology using sites).

The general idea of cohomology that I have at the moment is this: we could say that, at least in part,

modern geometry studies “spaces” by understanding **functions** on those spaces** **

where “space” could mean “variety”, “scheme”, “manifold”, “site”, and many other things. Cohomology is a way of understanding the functions on spaces in a pretty high-level way. It’s a way of going from local information about functions defined on neighbourhoods of points to global information about how we can extend those functions to global functions. The geometry of the “space” dictates whether functions can be “extended”, and cohomology is a way for us to unpack this information. Therefore we could say

cohomology is the study of **global properties** of a space by looking at **local-to-global obstructions.**

The words “local-to-global” should immediately bring to mind sheaves, and indeed cohomology basically seems to be the measure of losing exactness of sequences of sheaves

after taking global sections:

since the map on the right might not always be surjective. Sheaf cohomology seems to be the most general way (so far, to my understanding) of measuring these obstructions – and on suitably nice spaces, sheaf cohomology starts to agree with its more easily-calculable mates.

Anyway, I don’t want to talk so much about cohomology right now. Instead I wanted to briefly point out something cool on the Čech cohomology Wikipedia page. They have a picture there of a Penrose triangle:

By Tobias R. – Metoc – Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=2520370

Obviously this is the famous example of an “impossible object”: one that cannot exist in our 3D space. Things like this have delighted loads of people over the ages, and arguably great artists like M.C. Escher made careers out of exploiting these weird objects.

After spending ages working on cohomology in the abstract, I was amazed to hear that the “impossibility” of this Penrose triangle can be described by cohomology! This nice article by Penrose himself explains how the “impossibility” of the figure arises in the following way: we could chop up the figure into three pieces, each containing a corner of the triangle. By doing this we get three absolutely standard “corner pieces” that have no paradoxical qualities. The issue arises in **gluing** these pieces together, because we don’t have any information about the **distance** each piece is from the observer.

To explain what happens in the article, let’s fix two points such that when the corners are laid out as if to be glued, e.g. is superimposed over : in general and are the same to the observer except for the fact that they **may be a different distance away**. Let’s let denote the observer and

.

The ratios don’t depend on the actual distance the points are from the observer – only their ratios of the points with each other. By moving a one of the corners towards/away from the observer by a certain distance, we change the ratios and by the same factor . Therefore if the Penrose triangle could be realised as a 3D object it would be possible to simultaneously reduce each to 1 by transformations of this type. That would mean the **cocyle**

would be a **coboundary**, and would therefore be trivial in the first cohomology group of the annulus that surrounds the figure. But this is a nontrivial group, so not all cocycles are coboundaries, and it turns out this one is nontrivial.

So the nontriviality of the first cohomology group with real coefficients (distances from the observer) means that there are certain shapes, like the Penrose triangle, that cannot actually exist! The problem was that we had a shape (not the Penrose triangle, but the annulus surrounding it) and we had three open sets on it, each with a “corner” of the Penrose triangle drawn on it. Each of these open sets also had a function “distance from the observer” defined on it. The topology of the shape prevented us from being able to glue these functions together to get a globally consistent distance function. Which is precisely what causes the paradox with the Penrose triangle – at any point it is locally a consistent shape, but our brains cannot “glue” the distance information together to get a global shape that actually exists.

One small comment–perhaps entirely semantical. I think that your ordering of concepts is, perhaps, backwards from the way I think about things. Namely, I think that the high-level way of thinking about functions is sheaves, and that cohomology is just a way of quantifying (if one can think of groups as quantities) the difficulty of gluing local functions to global functions.

In fact, I think a way of making the statement “spaces are understood by their functions” can be made precise if one is willing to view functions under the highbrow lens of sheaves. Namely, one can say that if is any ‘sufficiently geometric’ category (schemes, varieties, manifolds, complex manifolds,…) then there is a natural embedding where is the category of locally ringed spaces. In other words, most sufficiently geometric categories can eschew their extra provided explicit structure by replacing it with functions–with a sheaf.

LikeLike

Also (I unfortunately can’t edit my comment) I think that the local-to-global phenomenon is much more apparent than the non right exactness of the global sections functor (even though that is, really, what is happening). Namely, if one looks at the definition of Cech cohomology it is *literally* measuring how difficult it is to find functions on individual opens which agree on overlaps. Since we’re working with sheaves, this is directly equivalent to asking ‘to what extent to local functions come from global functions?’

LikeLike

Dear Alex, thanks once again for some great comments. The idea that any sufficiently geometric category should embed into that of locally ringed spaces (and that the geometry of those spaces should be encoded in the sheaf of that locally ringed space, if that’s what you’re saying) is really nice. Essentially what I was hoping for with the idea of “understanding a space through its functions” although I wasn’t really sure how faithfully these categories could be embedded into that of locally ringed spaces. And, as you point out, considering sheaves as the “highbrow” extension of functions is the right thing to do, especially with *weird* spaces like schemes!

LikeLike