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. $\square$

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