Lemma 83.4.5 (049W)—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 March 12, 2021 at 10:45

for the reference: How about Tag 03JJ?

Comment #5973 by Laurent Moret-Bailly on March 12, 2021 at 11:52

What is proved here is more general: if $\phi$ factors through a monomorphism $i:X'\to X$ , then $i$ is an isomorphism.

Comment #5976 by Laurent Moret-Bailly on March 13, 2021 at 03:49

For what it's worth, here is a variant of the statement in my previous comment #5973: if a closed immersion $i:X'\to X$ of algebraic spaces is a categorical epimorphism (no "quotient" assumption involved) then it is an isomorphism. To prove this it suffices to show that $i$ is an equalizer. Now if $\mathcal{I}$ is the ideal sheaf of $X'$ , consider $\mathrm{Spec}(\oplus_{n\geq0}\mathcal{I}^n)\to X$ . This has two natural sections (the zero section, and the one deduced from $\mathcal{I}^n\subset\mathcal{O}_X$ ) whose equalizer is $X'$ .

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

Comment #5978 by Laurent Moret-Bailly on March 13, 2021 at 07:51

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 Johan on April 30, 2021 at 23:12

@#5972 Thanks and I added the reference here.

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

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