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In Atiyah-MacDonald's Commutative Algebra, they give in Exercise 16 of Chapter 1 the instruction:

Draw pictures of Spec($\mathbb{Z}$), Spec($\mathbb{R}$), Spec($\mathbb{C}[x]$), Spec($\mathbb{R}[x]$), Spec($\mathbb{Z}[x]$).

What exactly do they have in mind? I can enumerate the prime ideals in each of these rings, but I have little idea of what kinds of pictures one might draw and what advantage a good picture has over a mere enumeration.

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A good picture gives geometric intuition! There is an infamous picture of $\text{Spec}(\mathbb{Z}[x])$ which you can find here and which is probably the inspiration for this question: . – Qiaochu Yuan Jun 10 '12 at 3:07
You might take inspiration from the pictures in Vakil's notes. – Dylan Moreland Jun 10 '12 at 3:13
up vote 4 down vote accepted

Pictures are useful because they're suggestive, and are sometimes better at organizing information.

Sometimes the right picture is a little unclear. For example, when drawing $\mathop{\mathrm{Spec}} \mathbb{C}[x]$, sometimes you might want to draw a line, and imagine each complex number corresponds to some point on this line. Also, you want a splotch off to the side to represent the generic point of the line. This emphasizes the one-dimensionalness and the triviality of its structure as an algebraic curve, and also, it helps us to remember that it's ordinary topology doesn't play a role algebraically.

But other times, you would want to draw $\mathop{\mathrm{Spec}} \mathbb{C}[x]$ as the traditional complex plane with its ordinary labeling (again with a splotch to represent the generic point). This picture is useful, for example, if we want to apply results and intuition from complex analysis.

The main thing about having a picture for $\mathop{\mathrm{Spec}} \mathbb{Z}$ is, IMO, really just so that you can then draw a picture of $\mathop{\mathrm{Spec}} \mathbb{Z}[x]$ or to draw a picture of the spectrum of the ring of integers of a number field as a curve with a projection down to $\mathop{\mathrm{Spec}} \mathbb{Z}$.

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