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. $\square$

## Comments (0)

There are also:

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

## Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.

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 04T7. Beware of the difference between the letter 'O' and the digit '0'.