The Stacks project

Lemma 31.4.11. In Situation 31.4.5 we have the following:

  1. Given any quasi-compact open $V \subset S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$ there exists an $i \in I$ and a quasi-compact open $V_ i \subset S_ i$ such that $f_ i^{-1}(V_ i) = V$.

  2. Given $V_ i \subset S_ i$ and $V_{i'} \subset S_{i'}$ quasi-compact opens such that $f_ i^{-1}(V_ i) = f_{i'}^{-1}(V_{i'})$ there exists an index $i'' \geq i, i'$ such that $f_{i''i}^{-1}(V_ i) = f_{i''i'}^{-1}(V_{i'})$.

  3. If $V_{1, i}, \ldots , V_{n, i} \subset S_ i$ are quasi-compact opens and $S = f_ i^{-1}(V_{1, i}) \cup \ldots \cup f_ i^{-1}(V_{n, i})$ then $S_{i'} = f_{i'i}^{-1}(V_{1, i}) \cup \ldots \cup f_{i'i}^{-1}(V_{n, i})$ for some $i' \geq i$.

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 $i_0 = 0$. Choose an affine open covering $S_0 = U_{1, 0} \cup \ldots \cup U_{m, 0}$. Denote $U_{j, i} \subset S_ i$ the inverse image of $U_{j, 0}$ under the transition morphism for $i \geq 0$. Denote $U_ j$ the inverse image of $U_{j, 0}$ in $S$. Note that $U_ j = \mathop{\mathrm{lim}}\nolimits _ i U_{j, i}$ is a limit of affine schemes.

We first prove the uniqueness statement: Let $V_ i \subset S_ i$ and $V_{i'} \subset S_{i'}$ quasi-compact opens such that $f_ i^{-1}(V_ i) = f_{i'}^{-1}(V_{i'})$. It suffices to show that $f_{i''i}^{-1}(V_ i \cap U_{j, i''})$ and $f_{i''i'}^{-1}(V_{i'} \cap U_{j, i''})$ become equal for $i''$ large enough. Hence we reduce to the case of a limit of affine schemes. In this case write $S = \mathop{\mathrm{Spec}}(R)$ and $S_ i = \mathop{\mathrm{Spec}}(R_ i)$ for all $i \in I$. We may write $V_ i = S_ i \setminus V(h_1, \ldots , h_ m)$ and $V_{i'} = S_{i'} \setminus V(g_1, \ldots , g_ n)$. The assumption means that the ideals $\sum g_ jR$ and $\sum h_ jR$ have the same radical in $R$. This means that $g_ j^ N = \sum a_{jj'}h_{j'}$ and $h_ j^ N = \sum b_{jj'} g_{j'}$ for some $N \gg 0$ and $a_{jj'}$ and $b_{jj'}$ in $R$. Since $R = \mathop{\mathrm{colim}}\nolimits _ i R_ i$ we can chose an index $i'' \geq i$ such that the equations $g_ j^ N = \sum a_{jj'}h_{j'}$ and $h_ j^ N = \sum b_{jj'} g_{j'}$ hold in $R_{i''}$ for some $a_{jj'}$ and $b_{jj'}$ in $R_{i''}$. This implies that the ideals $\sum g_ jR_{i''}$ and $\sum h_ jR_{i''}$ have the same radical in $R_{i''}$ as desired.

We prove existence: If $S_0$ is affine, then $S_ i = \mathop{\mathrm{Spec}}(R_ i)$ for all $i \geq 0$ and $S = \mathop{\mathrm{Spec}}(R)$ with $R = \mathop{\mathrm{colim}}\nolimits R_ i$. Then $V = S \setminus V(g_1, \ldots , g_ n)$ for some $g_1, \ldots , g_ n \in R$. Choose any $i$ large enough so that each of the $g_ j$ comes from an element $g_{j, i} \in R_ i$ and take $V_ i = S_ i \setminus V(g_{1, i}, \ldots , g_{n, i})$. If $S_0$ is general, then the opens $V \cap U_ j$ are quasi-compact because $S$ is quasi-separated. Hence by the affine case we see that for each $j = 1, \ldots , m$ there exists an $i_ j \in I$ and a quasi-compact open $V_{i_ j} \subset U_{j, i_ j}$ whose inverse image in $U_ j$ is $V \cap U_ j$. Set $i = \max (i_1, \ldots , i_ m)$ and let $V_ i = \bigcup f_{ii_ j}^{-1}(V_{i_ j})$.

The statement on coverings follows from the uniqueness statement for the opens $V_{1, i} \cup \ldots \cup V_{n, i}$ and $S_ i$ of $S_ i$. $\square$


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