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 $U_0 = \bigcup _{j \in J} U_{0, j}$. 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 transtion 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 (0)


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Your email address will not be published.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 0CUG. Beware of the difference between the letter 'O' and the digit '0'.