Lemma 32.4.12. In Situation 32.4.5 if $S$ is quasi-affine, then for some $i_0 \in I$ the schemes $S_ i$ for $i \geq i_0$ are quasi-affine.

**Proof.**
Choose $i_0 \in I$. Note that $I$ is nonempty as the limit is directed. For convenience we write $S_0 = S_{i_0}$ and $i_0 = 0$. Let $s \in S$. We may choose an affine open $U_0 \subset S_0$ containing $f_0(s)$. Since $S$ is quasi-affine we may choose an element $a \in \Gamma (S, \mathcal{O}_ S)$ such that $s \in D(a) \subset f_0^{-1}(U_0)$, and such that $D(a)$ is affine. By Lemma 32.4.7 there exists an $i \geq 0$ such that $a$ comes from an element $a_ i \in \Gamma (S_ i, \mathcal{O}_{S_ i})$. For any index $j \geq i$ we denote $a_ j$ the image of $a_ i$ in the global sections of the structure sheaf of $S_ j$. Consider the opens $D(a_ j) \subset S_ j$ and $U_ j = f_{j0}^{-1}(U_0)$. Note that $U_ j$ is affine and $D(a_ j)$ is a quasi-compact open of $S_ j$, see Properties, Lemma 28.26.4 for example. Hence we may apply Lemma 32.4.11 to the opens $U_ j$ and $U_ j \cup D(a_ j)$ to conclude that $D(a_ j) \subset U_ j$ for some $j \geq i$. For such an index $j$ we see that $D(a_ j) \subset S_ j$ is an affine open (because $D(a_ j)$ is a standard affine open of the affine open $U_ j$) containing the image $f_ j(s)$.

We conclude that for every $s \in S$ there exist an index $i \in I$, and a global section $a \in \Gamma (S_ i, \mathcal{O}_{S_ i})$ such that $D(a) \subset S_ i$ is an affine open containing $f_ i(s)$. Because $S$ is quasi-compact we may choose a single index $i \in I$ and global sections $a_1, \ldots , a_ m \in \Gamma (S_ i, \mathcal{O}_{S_ i})$ such that each $D(a_ j) \subset S_ i$ is affine open and such that $f_ i : S \to S_ i$ has image contained in the union $W_ i = \bigcup _{j = 1, \ldots , m} D(a_ j)$. For $i' \geq i$ set $W_{i'} = f_{i'i}^{-1}(W_ i)$. Since $f_ i^{-1}(W_ i)$ is all of $S$ we see (by Lemma 32.4.11 again) that for a suitable $i' \geq i$ we have $S_{i'} = W_{i'}$. Thus we may replace $i$ by $i'$ and assume that $S_ i = \bigcup _{j = 1, \ldots , m} D(a_ j)$. This implies that $\mathcal{O}_{S_ i}$ is an ample invertible sheaf on $S_ i$ (see Properties, Definition 28.26.1) and hence that $S_ i$ is quasi-affine, see Properties, Lemma 28.27.1. Hence we win. $\square$

## 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Your email address will not be published.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.

## Comments (0)