Lemma 3.9.7. Let $S$ be a scheme. Let $f : X \to S$ be locally of finite type with $X$ quasi-compact. Then $\text{size}(X) \leq \text{size}(S)$.

Proof. We can find a finite affine open covering $X = \bigcup _{i = 1, \ldots n} U_ i$ such that each $U_ i$ maps into an affine open $S_ i$ of $S$. Thus by Lemma 3.9.5 we reduce to the case where both $S$ and $X$ are affine. In this case by Lemma 3.9.4 we see that it suffices to show

$|A[x_1, \ldots , x_ n]| \leq \max \{ \aleph _0, |A|\} .$

We omit the proof of this inequality. $\square$

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