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 formula_1 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.
Definition.
Motivation.
One of the motivating examples of an algebraic stack is to consider a groupoid scheme formula_2 over a fixed scheme formula_3. For example, if formula_4 (where formula_5 is the group scheme of roots of unity), formula_6, formula_7 is the projection map, formula_8 is the group actionformula_9and formula_10 is the multiplication mapformula_11on formula_5. Then, given an formula_3-scheme formula_14, the groupoid scheme formula_15 forms a groupoid (where formula_16 are their associated functors). Moreover, this construction is functorial on formula_17 forming a contravariant 2-functorformula_18where formula_19 is the 2-category of small categories. Another way to view this is as a fibred category formula_20 through the Grothendieck construction. Getting the correct technical conditions, such as the Grothendieck topology on formula_17, gives the definition of an algebraic stack. For instance, in the associated groupoid of formula_22-points for a field formula_22, over the origin object formula_24 there is the groupoid of automorphisms formula_25. Note that in order to get an algebraic stack from formula_26, and not just a stack, there are additional technical hypotheses required for formula_26.
Algebraic stacks.
It turns out using the fppf-topology (faithfully flat and locally of finite presentation) on formula_17, denoted formula_29, forms the basis for defining algebraic stacks. Then, an algebraic stack is a fibered categoryformula_30such that
Explanation of technical conditions.
Using the fppf topology.
First of all, the fppf-topology is used because it behaves well with respect to descent. For example, if there are schemes formula_37 and formula_38can be refined to an fppf-cover of formula_39, if formula_40 is flat, locally finite type, or locally of finite presentation, then formula_39 has this property. this kind of idea can be extended further by considering properties local either on the target or the source of a morphism formula_42. For a cover formula_43 we say a property formula_44 is local on the source ifformula_42 has formula_44 if and only if each formula_47 has formula_44.There is an analogous notion on the target called local on the target. This means given a cover formula_49formula_42 has formula_44 if and only if each formula_52 has formula_44.For the fppf topology, having an immersion is local on the target. In addition to the previous properties local on the source for the fppf topology, formula_54 being univerally 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 formula_55 is an fpqc-algebraic stackpg 40.
Representable diagonal.
By definition, a 1-morphism formula_56 of categories fibered in groupoids is representable by algebraic spaces meaning there exists an algebraic spaceformula_57such that the associated fibered category formula_58 is equivalent to formula_59. There are a number of equivalent conditions for representability of the diagonal which help give intuition for this technical condition, but one of main motivations is the following: for a scheme formula_60 and objects formula_61 the sheaf formula_62 is representable as an algebraic space. In particular, the stabilizer group for any point on the stack formula_63 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 productsformula_64the representability of the diagonal is equivalent to formula_65 being representable for an algebraic space formula_39. This is because given morphisms formula_67 from algebraic spaces, they extend to maps formula_68 from the diagonal map. There is an analogous statement for algebraic spaces which gives representability of a sheaf on formula_69 as an algebraic space.
Note that an analogous condition of representability of the diagonal holds for some formulations of higher stacks where the fiber product is an formula_70-stack for an formula_71-stack formula_31.
Surjective and smooth atlas.
2-Yoneda lemma.
The existence of an formula_34 scheme formula_35 and a 1-morphism of fibered categories formula_36 which is surjective and smooth depends on defining a smooth and surjective morphisms of fibered categories. Here formula_76 is the algebraic stack from the representable functor formula_77 on formula_78 upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the setformula_79is considered as a category, denoted formula_80, with objects in formula_81 as formula_34 morphismsformula_83and morphisms are the identity morphism. Henceformula_84is a 2-functor of groupoids. Showing this 2-functor is a sheaf is the content of the 2-Yoneda lemma. Using the Grothendieck construction, there is an associated category fibered in groupoids denoted formula_36.
Representable morphisms of categories fibered in groupoids.
To say this morphism formula_36 is smooth or surjective, we have to introduce representable morphisms. A morphism formula_87 of categories fibered in groupoids over formula_88 is said to be representable if given an object formula_89 in formula_88 and an object formula_91 the 2-fibered product formula_92is representable by a scheme. Then, we can say the morphism of categories fibered in groupoids formula_93 is smooth and surjective if the associated morphismformula_94of schemes is smooth and surjective.
Deligne-Mumford stacks.
Algebraic stacks, also known as Artin stacks, are by definition equipped with a smooth surjective atlas formula_36, where formula_76 is the stack associated to some scheme formula_35. If the atlas formula_98 is moreover étale, then formula_31 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 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 formula_100, the moduli stack of rank formula_71 vector bundles, has infinitesimal automorphisms controlled partially by the Lie algebra formula_102. 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 formula_103 is the deformation tractable, since the 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 formula_104. Also, the differential-geometric analogue of such stacks are called orbifolds. The Etale condition implies the 2-functorformula_105sending a scheme to its groupoid of formula_5-torsors is representable as a stack over the Etale topology, but the Picard-stack formula_107 of formula_108-torsors (equivalently the category of line bundles) is not representable. Stacks of this form are representable as stacks over the fppf-topology.
Another reason for considering the fppf-topology versus the etale topology is over characteristic formula_93 the Kummer sequenceformula_110is exact only as a sequence of fppf sheaves, but not as a sequence of etale sheaves.
Defining algebraic stacks over other topologies.
Using other Grothendieck topologies on formula_111 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 it useful to recall there is the following hierarchy of generalizationformula_112of big topologies on formula_111.
Structure sheaf.
The structure sheaf of an algebraic stack is an object pulled back from a universal structure sheaf formula_114 on the site formula_88. This universal structure sheaf is defined asformula_116and the associated structure sheaf on a category fibered in groupoidsformula_117is defined asformula_118where formula_119 comes from the map of Grothendieck topologies. In particular, this means is formula_120 lies over formula_60, so formula_122, then formula_123. As a sanity check, it's worth comparing this to a category fibered in groupoids coming from an formula_3-scheme formula_40 for various topologies. For example, if formula_126is a category fibered in groupoids over formula_88, the structure sheaf for an open subscheme formula_128 givesformula_129so this definition recovers the classic structure sheaf on a scheme. Moreover, for a quotient stack formula_130, the structure sheaf this just gives the formula_131-invariant sectionsformula_132for formula_133 in formula_88.
Examples.
Classifying stacks.
Many classifying stacks for algebraic groups are algebraic stacks. In fact, for an algebraic group space formula_131 over a scheme formula_3 which is flat of finite presentation, the stack formula_137 is algebraictheorem 6.1.