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Atiyah/Macdonald's commutative algebra book asks the reader to draw pictures of the prime spectrum of $\mathbb{Z}$ in exercise 1.16. I worked through it on my own, figured out what the space looks like as a set and where the open/closed sets are, but when I did some research to check my work I found the following picture:enter image description here

What exactly is the squiggly line supposed to represent? is it meant to be the ideal $(0)$? That makes a vague bit of sense to me, as the closure of (0) is the entire space, but it contradicts the fact that (0) is a single point in the prime spectrum. Another source said to draw $\operatorname{Spec}(\mathbb{Z})$ as a curve passing through the ideals $(2),(3),(5),\ldots$ and a separate point off to the side to represent $(0)$, but again I don't get how drawing a curve helps clarify the structure of the space. Why not draw it as a discrete set of points and leave out the curve?

Edit: Here's another guess I just came up with: If we put the finite-complement topology on the real line and then create a quotient space by identifying every point that is not a prime positive integer as one point and call it (0), do we get $\operatorname{Spec}(\mathbb{Z})$? If this is the case it would definitely help justify the drawing.

Nick A.
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2 Answers2

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$(0)$ is the generic point. It is a single point, in some sense, "nowhere in particular" (quoting Vakil). A way to think of it is that closed points are actual points, and generic points are "fat/fuzzy" points (quoting here)

The generic point is drawn as going through all of the other points because the closure of the generic point is the entire space.

David Lui
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  • Thanks! This makes a lot of sense. Was I right that the quotient space I described and the Zariski topology on Spec(Z) coincide? I'm trying to come up with a few different angles to understand the space. – Nick A. Feb 02 '21 at 00:21
  • Yes I think so, since the topology on $Spec(\mathbb{Z})$ is the intersection of the cofinite topology and the particular point topology. – David Lui Feb 02 '21 at 00:32
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I really like Yuri Manin's way of describing how points and generic points build up affine schemes. In this sense curves in $\mathbb{C}[T_1, C_2]$ have closed points but also generic points that sit on a "lower level". These generic points in turn sit above the generic point $(0)$.

For $\operatorname{Spec} \mathbb{Z}$ it would be points sitting on a plane with the generic point below them, pretty much the two lower levels of the picture below.

$\hskip2in$C

Lejoon
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