↑ arvau

Wikipedia / 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 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 Alexander 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. After Grothendieck developed the general theory of descent, and Giraud the general theory of stacks, the notion of algebraic stacks was defined by Michael Artin.

Definition

Motivation

One of the motivating examples of an algebraic stack is to consider a groupoid scheme is the group scheme of roots of unity is the group action is the multiplication map forming a contravariant 2-functor is the 2-category of small categories. Another way to view this is as a fibred category through the Grothendieck construction. Getting the correct technical conditions, such as the Grothendieck topology.

Algebraic stacks

It turns out using the fppf-topology (faithfully flat and locally of finite presentation, forms the basis for defining algebraic stacks. Then, an algebraic stack is a fibered categorysuch that

  1. is a category fibered in groupoids, meaning the overcategory is a groupoid

  2. of fibered categories is representable as algebraic spaces

  3. which is surjective and smooth called an atlas.

Explanation of technical conditions

Using the fppf topology

First of all, the fppf-topology is used because it behaves well with respect to descent has this property. is local on the source if.There is an analogous notion on the target called local on the target.For the fppf topology, having an immersion is local on the target. being universally open is also local on the source. Also, being locally Noetherian and Jacobson are local on the source and target for the fppf topology. 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 is an fpqc-algebraic stackpg 40.

Representable diagonal

of categories fibered in groupoids is representable by algebraic spaces, the associated category fibered in groupoidsis representable as an algebraic space, meaning there exists an algebraic space. There are a number of equivalent conditions for representability of the diagonal is representable as an algebraic space.

Another important equivalence of having a representable diagonal is the technical condition that the intersection of any two algebraic spaces in an algebraic stack is an algebraic space. Reformulated using fiber products as an algebraic space.

Note that an analogous condition of representability of the diagonal holds for some formulations of higher stacks.

Surjective and smooth atlas

2-Yoneda lemma

is the algebraic stack from the representable functor upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the set morphismsand morphisms are the identity morphism. Henceis a 2-functor of groupoids. Showing this 2-functor is a sheaf is the content of the 2-Yoneda lemma.

Representable morphisms of categories fibered in groupoids

is smooth or surjective, we have to introduce representable morphisms. the 2-fibered product is smooth and surjective if the associated morphismof schemes is smooth and surjective.

Deligne–Mumford stacks

Algebraic stacks, also known as Artin stacks is said to be a Deligne–Mumford stack. The subclass of Deligne-Mumford stacks is useful because it provides the correct setting for many natural stacks considered, such as the moduli stack of algebraic curves. In addition, they are strict enough that objects represented by points in Deligne-Mumford stacks do not have infinitesimal automorphisms 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 many stacks cannot be naturally represented as Deligne-Mumford stacks because it only allows for finite covers, or, algebraic stacks with finite covers. 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. The Etale condition implies the 2-functor-torsors-torsors (equivalently the category of line bundles) is not representable. Stacks of this form are representable as stacks over the fppf-topology.

the Kummer sequenceis exact only as a sequence of fppf sheaves, but not as a sequence of etale sheaves.

Defining algebraic stacks over other topologies

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. It is useful to recall there is the following hierarchy of generalization.

Structure sheaf

. This universal structure sheaf is defined asand the associated structure sheaf on a category fibered in groupoidsis defined as for various topologies. For example, if givesso this definition recovers the classic structure sheaf on a scheme. Moreover, for a quotient stack-invariant sections.

Examples

Classifying stacks

Many classifying stacks for algebraic groups is algebraictheorem 6.1.

See also

References

Artin's Axioms

Papers

Applications

Other