The Stacks project

Lemma 70.4.7. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. There are canonical bijections between the following sets:

  1. the set of points of $X$, i.e., $|X|$,

  2. the set of irreducible closed subsets of $|X|$,

  3. the set of integral closed subspaces of $X$.

The bijection from (1) to (2) sends $x$ to $\overline{\{ x\} }$. The bijection from (3) to (2) sends $Z$ to $|Z|$.

Proof. Our map defines a bijection between (1) and (2) as $|X|$ is sober by Decent Spaces, Proposition 66.12.4. Given $T \subset |X|$ closed and irreducible, there is a unique reduced closed subspace $Z \subset X$ such that $|Z| = T$, namely, $Z$ is the reduced induced subspace structure on $T$, see Properties of Spaces, Definition 64.12.5. This is an integral algebraic space because it is decent, reduced, and irreducible. $\square$


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