The Stacks project

Lemma 32.4.4. Let $S = \mathop{\mathrm{lim}}\nolimits S_ i$ be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma 32.2.2). Let $s \in S$ with images $s_ i \in S_ i$. Then

  1. $s = \mathop{\mathrm{lim}}\nolimits s_ i$ as schemes, i.e., $\kappa (s) = \mathop{\mathrm{colim}}\nolimits \kappa (s_ i)$,

  2. $\overline{\{ s\} } = \mathop{\mathrm{lim}}\nolimits \overline{\{ s_ i\} }$ as sets, and

  3. $\overline{\{ s\} } = \mathop{\mathrm{lim}}\nolimits \overline{\{ s_ i\} }$ as schemes where $\overline{\{ s\} }$ and $\overline{\{ s_ i\} }$ are endowed with the reduced induced scheme structure.

Proof. Choose $0 \in I$ and an affine open covering $S_0 = \bigcup _{j \in J} U_{0, j}$. For $i \geq 0$ let $U_{i, j} = f_{i, 0}^{-1}(U_{0, j})$ and set $U_ j = f_0^{-1}(U_{0, j})$. Here $f_{i'i} : S_{i'} \to S_ i$ is the transition morphism and $f_ i : S \to S_ i$ is the projection. For $j \in J$ the following are equivalent: (a) $s \in U_ j$, (b) $s_0 \in U_{0, j}$, (c) $s_ i \in U_{i, j}$ for all $i \geq 0$. Let $J' \subset J$ be the set of indices for which (a), (b), (c) are true. Then $\overline{\{ s\} } = \bigcup _{j \in J'} (\overline{\{ s\} } \cap U_ j)$ and similarly for $\overline{\{ s_ i\} }$ for $i \geq 0$. Note that $\overline{\{ s\} } \cap U_ j$ is the closure of the set $\{ s\} $ in the topological space $U_ j$. Similarly for $\overline{\{ s_ i\} } \cap U_{i, j}$ for $i \geq 0$. Hence it suffices to prove the lemma in the case $S$ and $S_ i$ affine for all $i$. This reduces us to the algebra question considered in the next paragraph.

Suppose given a system of rings $(A_ i, \varphi _{ii'})$ over $I$. Set $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$ with canonical maps $\varphi _ i : A_ i \to A$. Let $\mathfrak p \subset A$ be a prime and set $\mathfrak p_ i = \varphi _ i^{-1}(\mathfrak p)$. Then

\[ V(\mathfrak p) = \mathop{\mathrm{lim}}\nolimits _ i V(\mathfrak p_ i) \]

This follows from Lemma 32.4.1 because $A/\mathfrak p = \mathop{\mathrm{colim}}\nolimits A_ i/\mathfrak p_ i$. This equality of rings also shows the final statement about reduced induced scheme structures holds true. The equality $\kappa (\mathfrak p) = \mathop{\mathrm{colim}}\nolimits \kappa (\mathfrak p_ i)$ follows from the statement as well. $\square$

Comments (2)

Comment #6660 by Jonas Ehrhard on

Should it be in the first sentence of the proof?

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