Lemma 3.9.11. Let $f : X \to Y$ be a morphism of schemes. Assume there exists an fpqc covering $\{ g_ j : Y_ j \to Y\} _{j \in J}$ such that $g_ j$ factors through $f$. Then $\text{size}(Y) \leq \text{size}(X)$.

Proof. Let $V \subset Y$ be an affine open. By definition there exist $n \geq 0$ and $a : \{ 1, \ldots , n\} \to J$ and affine opens $V_ i \subset Y_{a(i)}$ such that $V = g_{a(1)}(V_1) \cup \ldots \cup g_{a(n)}(V_ n)$. Denote $h_ j : Y_ j \to X$ a morphism such that $f \circ h_ j = g_ j$. Then $h_{a(1)}(V_1) \cup \ldots \cup h_{a(n)}(V_ n)$ is a quasi-compact subset of $f^{-1}(V)$. Hence we can find a quasi-compact open $W \subset f^{-1}(V)$ which contains $h_{a(i)}(V_ i)$ for $i = 1, \ldots , n$. In particular $V = f(W)$.

On the one hand this shows that the cardinality of the set of affine opens of $Y$ is at most the cardinality of the set $S$ of quasi-compact opens of $X$. Since every quasi-compact open of $X$ is a finite union of affines, we see that the cardinality of this set is at most $\sup |S|^ n = \max (\aleph _0, |S|)$. On the other hand, we have $\mathcal{O}_ Y(V) \subset \prod _{i = 1, \ldots , n} \mathcal{O}_{Y_{a(i)}}(V_ i)$ because $\{ V_ i \to V\}$ is an fpqc covering. Hence $\mathcal{O}_ Y(V) \subset \mathcal{O}_ X(W)$ because $V_ i \to V$ factors through $W$. Again since $W$ has a finite covering by affine opens of $X$ we conclude that $|\mathcal{O}_ Y(V)|$ is bounded by the size of $X$. The lemma now follows from the definition of the size of a scheme. $\square$

There are also:

• 4 comment(s) on Section 3.9: Constructing categories of schemes

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AHK. Beware of the difference between the letter 'O' and the digit '0'.