Lemma 26.21.5 (01KM)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 26: Schemes
Section 26.21: Separation axioms
Lemma 26.21.5 (cite)

numberstags

Lemma 26.21.5. Let $X$ , $Y$ be schemes over $S$ . Let $a, b : X \to Y$ be morphisms of schemes over $S$ . There exists a largest locally closed subscheme $Z \subset X$ such that $a|_ Z = b|_ Z$ . In fact $Z$ is the equalizer of $(a, b)$ . Moreover, if $Y$ is separated over $S$ , then $Z$ is a closed subscheme.

Proof. The equalizer of $(a, b)$ is for categorical reasons the fibre product $Z$ in the following diagram

$\xymatrix{ Z = Y \times _{(Y \times _ S Y)} X \ar[r] \ar[d] & X \ar[d]^{(a , b)} \\ Y \ar[r]^-{\Delta _{Y/S}} & Y \times _ S Y }$

Thus the lemma follows from Lemmas 26.18.2, 26.21.2 and Definition 26.21.3. $\square$

Comments (2)

Comment #7077 by Zeyn Sahilliogullari on February 24, 2022 at 18:09

Is "locally closed subscheme" explicitly defined elsewhere in the stacks project?

Comment #7089 by Zeyn Sahilliogullari on March 03, 2022 at 16:54

@#7077 "locally closed subscheme" is defined at the end of the section on scheme immersions:
https://stacks.math.columbia.edu/tag/01IM

There are also:

19 comment(s) on Section 26.21: Separation axioms

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 (2)

Post a comment