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The trick is to think of a 'number ring' like

$\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5}:a,b\in \mathbb{Z}\}$

as the ring of functions on a kind of space, called a 'scheme'. Then ideals in your number ring act like line bundles on this space!

You can multiply a section of a line bundle by a function; similarly you can multiply an element of an ideal by an element of your number ring. That's how the analogy starts.

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Carrying it out much further, people developed 'class field theory', which applies cohomology to number theory. I've found Jürgen Neukirch's book Algebraic Number Theory to be one of the most fun ways to learn about this stuff. His love for the subject shines through.

Let me explain one tiny idea: how the ideal class group of a number ring is a cohomology group.

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An ideal I of a number ring K gives a module of K, just as a line bundle over a topological space gives a module for the ring of continuous functions on that space.

In both cases we don't get just any old module, either: we get a locally free module of rank 1. This is an algebraic way of saying that 'locally', our module looks just like a copy of the ring it's a module of. But 'globally' it can twist around, like our pal the Möbius strip.

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Now, some line bundles on a space are 'trivial'. These give modules of its ring of functions that don't twist around at all! These modules are not just 'locally free of rank 1' but *actually* free of rank 1. This is a fancy way to say they're isomorphic to the ring itself.

Similarly for ideals of a number ring K. Some are actually free of rank 1 - that is, isomorphic to K itself. These are the 'principal' ideals:

https://en.wikipedia.org/wiki/Principal_ideal

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The analogy continuues: we can multiply line bundles, and also multiply ideals.

To multiply or 'tensor' two line bundles L and L' we take the tensor product of their fibers, which are 1d vector spaces, and get a new 1d vector space which we take as the fiber of a new line bundle L⊗L'.

To multiply two ideals I and I' of a number ring we simply multiply elements of I with elements of I' and take all possible sums, getting an ideal II'.

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When we multiply line bundles their twistiness gets added! This suggests that we should form an abelian group from line bundles, or some equivalence classes of line bundles. We can do it: just take isomorphism classes of line bundles and make it into a group using

[L] [L'] = [L⊗L']

We can do something similar with ideals of a number ring, but it's a bit subtler. We get a group called the 'ideal class group':

https://en.wikipedia.org/wiki/Ideal_class_group

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The subtlety in the number theory case is that not every locally free module of rank 1 comes from an ideal: we need general things called 'fractional ideals'. But if just talk about modules this subtlety disappears.

So: in both topology and number theory, there's an important abelian group whose elements are the ways that a line bundle (or locally free module of rank 1) can be twisted!

And some good news: in both cases, this is a cohomology group!

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If X is a topological space, its isomorphism classes of complex line bundles correspond to elements of the group H¹(X,ℂ*). This is the first cohomology of X with coefficients in ℂ*, the group of invertible complex numbers!

If K is a number ring, its ideal class group is isomorphic to H¹(K,𝔾ₘ). This is the first cohomology of K with coefficients in 𝔾ₘ.

What's 𝔾ₘ? It stands for 'multiplicative group', and it's analogous to ℂ*. But here it's a sheaf:

https://math.stackexchange.com/questions/1971630/what-does-mathbf-g-m-really-mean

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So: the ideal class group of a number field really is a first cohomology group H¹(K,𝔾ₘ).

And *morally*, not literally, it's first cohomology because it describes how something like a line bundle twists as we march around a 1-dimensional loop in some space - a scheme - and it has coefficients in 𝔾ₘ because the 'twist' is an invertible number of some sort.

This is so cool. It means you can use your topological intuition in number theory!

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@johncarlosbaez I really appreciate the effort you're putting into building a community here.
Could you please try to make sure to add captions to images you post? When it's a screenshot of a block of text, you can use the "detect text from picture" button in the web app to have it transcribed for you.