Quotient space of an algebraic stack explained

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.^[1]

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, 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).

Notes and References

1. 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| .