The Stacks project

Lemma 32.4.1. 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). Then $S_{set} = \mathop{\mathrm{lim}}\nolimits _ i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$.

Proof. Pick $i \in I$. Take $U_ i \subset S_ i$ an affine open. Denote $U_{i'} = f_{i'i}^{-1}(U_ i)$ and $U = f_ i^{-1}(U_ i)$. Here $f_{i'i} : S_{i'} \to S_ i$ is the transtion morphism and $f_ i : S \to S_ i$ is the projection. By Lemma 32.2.2 we have $U = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} U_ i$. Suppose we can show that $U_{set} = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} U_{i', set}$. Then the lemma follows by a simple argument using an affine covering of $S_ i$. Hence we may assume all $S_ i$ and $S$ affine. 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 $\varphi _ i : A_ i \to A$. Then

\[ \mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{Spec}}(A_ i) \]

Namely, suppose that we are given primes $\mathfrak p_ i \subset A_ i$ such that $\mathfrak p_ i = \varphi _{ii'}^{-1}(\mathfrak p_{i'})$ for all $i' \geq i$. Then we simply set

\[ \mathfrak p = \{ x \in A \mid \exists i, x_ i \in \mathfrak p_ i \text{ with }\varphi _ i(x_ i) = x\} \]

It is clear that this is an ideal and has the property that $\varphi _ i^{-1}(\mathfrak p) = \mathfrak p_ i$. Then it follows easily that it is a prime ideal 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Your email address will not be published.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 0CUE. Beware of the difference between the letter 'O' and the digit '0'.