The Stacks project

Lemma 83.4.5. In the situation of Definition 83.4.1. If $\phi : U \to X$ is a categorical quotient and $U$ is reduced, then $X$ is reduced. The same holds for categorical quotients in a category of spaces $\mathcal{C}$ listed in Example 83.4.3.

Proof. Let $X_{red}$ be the reduction of the algebraic space $X$. Since $U$ is reduced the morphism $\phi : U \to X$ factors through $i : X_{red} \to X$ (Properties of Spaces, Lemma 66.12.4). Denote this morphism by $\phi _{red} : U \to X_{red}$. Since $\phi \circ s = \phi \circ t$ we see that also $\phi _{red} \circ s = \phi _{red} \circ t$ (as $i : X_{red} \to X$ is a monomorphism). Hence by the universal property of $\phi $ there exists a morphism $\chi : X \to X_{red}$ such that $\phi _{red} = \phi \circ \chi $. By uniqueness we see that $i \circ \chi = \text{id}_ X$ and $\chi \circ i = \text{id}_{X_{red}}$. Hence $i$ is an isomorphism and $X$ is reduced.

To show that this argument works in a category $\mathcal{C}$ one just needs to show that the reduction of an object of $\mathcal{C}$ is an object of $\mathcal{C}$. We omit the verification that this holds for each of the standard examples. $\square$


Comments (5)

Comment #5972 by Dario Weißmann on

for the reference: How about Tag 03JJ?

Comment #5973 by Laurent Moret-Bailly on

What is proved here is more general: if factors through a monomorphism , then is an isomorphism.

Comment #5976 by Laurent Moret-Bailly on

For what it's worth, here is a variant of the statement in my previous comment #5973: if a closed immersion of algebraic spaces is a categorical epimorphism (no "quotient" assumption involved) then it is an isomorphism. To prove this it suffices to show that is an equalizer. Now if is the ideal sheaf of , consider . This has two natural sections (the zero section, and the one deduced from ) whose equalizer is .

You can weaken "closed immersion" to "(quasiccompact) immersion" if you restrict to algebraic spaces with separation conditions.

Comment #5978 by Laurent Moret-Bailly on

Sorry, the last paragraph of my comment #5976 is of course nonsense: you may allow general immersions but then you cannot restrict to separated spaces!

Comment #6151 by on

@#5972 Thanks and I added the reference here.

Dear Laurent, that is a really nice argument! Best, Johan


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Your email address will not be published.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 049W. Beware of the difference between the letter 'O' and the digit '0'.