In algebraic geometry, the quotient space of an

algebraic stack In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's repr ...

''F'', denoted by , ''F'', , is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

which as a set is the set of all integral substacks of ''F'' and which then is given a "

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...

": 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 Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

. When ''X'' is a moduli stack, the quotient space

, X,

is called the

moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...

of ''X''. If

f: X \to Y

is a morphism of algebraic stacks that induces a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

f: , X,  \overset\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). {{algebraic-geometry-stub Algebraic geometry