Quotient space of an algebraic stack

In algebraic geometry, the quotient space of an algebraic stack F, denoted by |F|, is a topological space which as a set is the set of all integral substacks of F and which then is given a "Zariski topology": an open subset has a form $|U| \subset |F|$ for some open substack U of F.In other words, there is a natural bijection between the set of all open immersions to F and the set of all open subsets of $|F|$ .

The construction $X \mapsto |X|$ is functorial; i.e., each morphism $f: X \to Y$ of algebraic stacks determines a continuous map $f: |X| \to |Y|$ .

An algebraic stack X is punctual if $|X|$ is a point.

When X is a moduli stack, the quotient space $|X|$ is called the moduli space of X. If $f: X \to Y$ is a morphism of algebraic stacks that induces a homeomorphism $f: |X| \overset{\sim}\to |Y|$ , then Y is called a coarse moduli stack of X. ("The" coarse moduli requires a universality.)

References

H. Gillet, [https://www.sciencedirect.com/science/article/pii/0022404984900367 Intersection theory on algebraic stacks and Q-varieties], J. Pure Appl. Algebra 34 (1984), 193–240, Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983).

Category:Algebraic geometry