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
After Grothendieck developed the general theory of descent,[2] and Giraud the general theory of stacks,[3] the notion of algebraic stacks was defined by Michael Artin.
One of the motivating examples of an algebraic stack is to consider a groupoid scheme
It turns out using the fppf-topology[6] (faithfully flat and locally of finite presentation) on
[8] this kind of idea can be extended further by considering properties local either on the target or the source of a morphism
[9] In addition to the previous properties local on the source for the fppf topology,
[11] This does not hold in the fpqc topology, making it not as "nice" in terms of technical properties.
Even though this is true, using algebraic stacks over the fpqc topology still has its use, such as in chromatic homotopy theory.
This is because the Moduli stack of formal group laws
of categories fibered in groupoids is representable by algebraic spaces[13] if for any fppf morphism
There is an analogous statement for algebraic spaces which gives representability of a sheaf on
[18] Note that an analogous condition of representability of the diagonal holds for some formulations of higher stacks[19] where the fiber product is an
Using the Grothendieck construction, there is an associated category fibered in groupoids denoted
is smooth or surjective, we have to introduce representable morphisms.
In addition, they are strict enough that object represented by points in Deligne-Mumford stacks do not have infinitesimal automorphisms.
This is very important because infinitesimal automorphisms make studying the deformation theory of Artin stacks very difficult.
For example, the deformation theory of the Artin stack
vector bundles, has infinitesimal automorphisms controlled partially by the Lie algebra
This leads to an infinite sequence of deformations and obstructions in general, which is one of the motivations for studying moduli of stable bundles.
Only in the special case of the deformation theory of line bundles
is the deformation theory tractable, since the associated Lie algebra is abelian.
Note that because every Etale cover is flat and locally of finite presentation, algebraic stacks defined with the fppf-topology subsume this theory; but, it is still useful since many stacks found in nature are of this form, such as the moduli of curves
Also, the differential-geometric analogue of such stacks are called orbifolds.
-torsors is representable as a stack over the Etale topology, but the Picard-stack
-torsors (equivalently the category of line bundles) is not representable.
Another reason for considering the fppf-topology versus the etale topology is over characteristic
gives alternative theories of algebraic stacks which are either not general enough, or don't behave well with respect to exchanging properties from the base of a cover to the total space of a cover.
and the associated structure sheaf on a category fibered in groupoids
As a sanity check, it's worth comparing this to a category fibered in groupoids coming from an
so this definition recovers the classic structure sheaf on a scheme.