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Concept
Algebraic stack
Summary
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 representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin.
One of the motivating examples of an algebraic stack is to consider a groupoid scheme over a fixed scheme . For example, if (where is the group scheme of roots of unity), , is the projection map, is the group actionand is the multiplication mapon . Then, given an -scheme , the groupoid scheme forms a groupoid (where are their associated functors). Moreover, this construction is functorial on forming a contravariant 2-functorwhere is the of . Another way to view this is as a through the Grothendieck construction. Getting the correct technical conditions, such as the Grothendieck topology on , gives the definition of an algebraic stack. For instance, in the associated groupoid of -points for a field , over the origin object there is the groupoid of automorphisms . Note that in order to get an algebraic stack from , and not just a stack, there are additional technical hypotheses required for .
It turns out using the fppf-topology (faithfully flat and locally of finite presentation) on , denoted , forms the basis for defining algebraic stacks. Then, an algebraic stack is a fibered categorysuch that
is a , meaning the for some is a groupoid
The diagonal map of fibered categories is representable as algebraic spaces
There exists an scheme and an associated 1-morphism of fibered categories which is surjective and smooth called an atlas.
Official source
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