Quasi-compactness of the spectrum

Remarks

It is enough to treat covers by basic open sets $D(f_i) = \set{\mathfrak p : f_i \notin \mathfrak p}$, $i \in I$. These cover $\operatorname{Spec} R$ exactly when no prime ideal contains all $f_i$. Since every proper ideal lies in a maximal ideal, this means that the $f_i$ generate the unit ideal. Then $1 = a_1 f_{i_1} + \cdots + a_n f_{i_n}$ for finitely many indices, and $D(f_{i_1}), \ldots, D(f_{i_n})$ already cover.

Example: $D(2)$ and $D(3)$ cover $\operatorname{Spec} \Z$, because $1 = 3 - 2$.

The word "quasi-compact" is used because the space is usually not Hausdorff, and some authors reserve "compact" for Hausdorff spaces. The same argument shows that each basic open set $D(f)$ is quasi-compact, since $D(f) \cong \operatorname{Spec} R_f$.

Going deeper

For a student

Quasi-compactness of $\operatorname{Spec} R$ is the algebraic shadow of the fact that $1$ is a finite sum. It is used to pass from local to global data in finitely many steps. For example, it lets one glue finitely many local constructions, or bound a quantity over a finite cover. In scheme theory, it says that affine schemes are quasi-compact. It is one of the conditions in Hochster's theorem on spectral spaces.