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So, I'm looking into schemes, and found that I have no intuition in the field, so I decided to look into some simple (as in affine and well-known) examples. As I like to dwell on the basics for a while, and texts on graduate level tend to move too quickly away from the basics for me, there isn't much material. These are some of the conclusions I've come to so far:

First of all, the Zarisky topology on Spec$(\mathbb Z)$ has as closed sets any finite set not containing $(0)$, as well as the whole set.

Second, let the open set $U$ be the complement of the union of the prime ideals generated by the primes $p_1, \ldots, p_n$, in other words, $U$ consists of all the prime ideals not containing the product $p_1p_2\cdots p_n$. Then the sheaf over Spec$(\mathbb Z)$ takes $U$ to the localization $\mathbb Z_{p_1p_2\cdots p_n}$, i.e. the subring of $\mathbb Q$ consisting of rationals which can be written as a fraction with the denominator a power of $p_1p_2\cdots p_n$.

Third, the stalk around a prime ideal $(p)$ is $\mathbb Z_{(p)}$, that is, the subring of $\mathbb Q$ consisting of rationals that can be written as a fraction without any factor $p$ in the denominator. As a special case, the stalk around $(0)$ is $\mathbb Q$.

Am I wrong about any of this?

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This all looks good to me — the last two assertions are special cases of the standard propositions on $\mathscr{O}(D(f))$s and $\mathscr{O}_\mathfrak{p}$s for affine schemes. Have you tried reading Vakil's notes? They move quickly, but there are examples at every turn. –  Dylan Moreland Jun 19 '12 at 18:25
Eisenbud and Harris's Geometry of schemes spends quite a few pages on $\operatorname{Spec} \mathbb{Z}$. Perhaps you'd like to take a look at that. What you say is more or less correct. –  Zhen Lin Jun 19 '12 at 18:37
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up vote 1 down vote accepted

Turns out it was all good, according to the two commments. I even got a few pointers to material, which was a nice bonus. Thanks a lot, Zhen and Dylan.

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