Lemma 10.29.5 (00F9)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.29: Images of ring maps of finite presentation
Lemma 10.29.5 (cite)

numberstags

Lemma 10.29.5. Let $R$ be a ring. Let $f$ be an element of $R$ . Let $S = R_ f$ . Then the image of a constructible subset of $\mathop{\mathrm{Spec}}(S)$ is constructible in $\mathop{\mathrm{Spec}}(R)$ .

Proof. We repeatedly use Lemma 10.29.1 without mention. Let $U, V$ be quasi-compact open in $\mathop{\mathrm{Spec}}(S)$ . We will show that the image of $U \cap V^ c$ is constructible. Under the identification $\mathop{\mathrm{Spec}}(S) = D(f)$ of Lemma 10.17.6 the sets $U, V$ correspond to quasi-compact opens $U', V'$ of $\mathop{\mathrm{Spec}}(R)$ . Hence it suffices to show that $U' \cap (V')^ c$ is constructible in $\mathop{\mathrm{Spec}}(R)$ which is clear. $\square$

Comments (0)

There are also:

6 comment(s) on Section 10.29: Images of ring maps of finite presentation

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).

Comments (0)

Post a comment