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 [math]\displaystyle{ |U| \subset |F| }[/math] for some open substack U of F.^[1] The construction [math]\displaystyle{ X \mapsto |X| }[/math] is functorial; i.e., each morphism [math]\displaystyle{ f: X \to Y }[/math] of algebraic stacks determines a continuous map [math]\displaystyle{ f: |X| \to |Y| }[/math].

An algebraic stack X is punctual if [math]\displaystyle{ |X| }[/math] is a point.

When X is a moduli stack, the quotient space [math]\displaystyle{ |X| }[/math] is called the moduli space of X. If [math]\displaystyle{ f: X \to Y }[/math] is a morphism of algebraic stacks that induces a homeomorphism [math]\displaystyle{ f: |X| \overset{\sim}\to |Y| }[/math], 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).

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