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Archive for the ‘arithmetic geometry’ Category

If you haven’t seen them already, you might want to read John Baez’s week205 and Lieven le Bruyn’s series of posts on the subject of spectra. I especially recommend that you take a look at the picture of \text{Spec } \mathbb{Z}[x] to which Lieven le Bruyn links before reading this post. John Baez’s introduction to week205 would probably also have served as a great introduction to this series before I started it:

There’s a widespread impression that number theory is about numbers, but I’d like to correct this, or at least supplement it. A large part of number theory – and by the far the coolest part, in my opinion – is about a strange sort of geometry. I don’t understand it very well, but that won’t prevent me from taking a crack at trying to explain it….

Before we talk about localization again, we need some examples of rings to localize. Recall that our proof of the description of \text{Spec } \mathbb{C}[x, y] also gives us a description of \text{Spec } \mathbb{Z}[x]:

Theorem: \text{Spec } \mathbb{Z}[x] consists of the ideals (0), (f(x)) where f is irreducible, and the maximal ideals (p, f(x)) where p \in \mathbb{Z} is prime and f(x) is irreducible in \mathbb{F}_p[x].

The upshot is that we can think of the set of primes of a ring of integers \mathbb{Z}[\alpha] \simeq \mathbb{Z}[x]/(f(x)), where f(x) is a monic irreducible polynomial with integer coefficients, as an “algebraic curve” living in the “plane” \text{Spec } \mathbb{Z}[x], which is exactly what we’ll be doing today. (When f isn’t monic, unfortunate things happen which we’ll discuss later.) We’ll then cover the case of actual algebraic curves next.

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