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Algebraic stack

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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[1] 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,[2] and Giraud the general theory of stacks,[3] the notion of algebraic stacks was defined by Michael Artin.[4]

Definition

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Motivation

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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 action

and is the multiplication map

on . 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-functor

where 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 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 . However, in order to get an algebraic stack from , and not just a stack, there are additional technical hypotheses required for .[5]

Algebraic stacks

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It turns out using the fppf-topology[6] (faithfully flat and locally of finite presentation) on , denoted , forms the basis for defining algebraic stacks. Then, an algebraic stack[7] is a fibered category

such that

  1. is a category fibered in groupoids, meaning the overcategory for some is a groupoid
  2. The diagonal map of fibered categories is representable as algebraic spaces
  3. There exists an scheme and an associated 1-morphism of fibered categories which is surjective and smooth called an atlas.

Explanation of technical conditions

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Using the fppf topology
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First of all, the fppf-topology is used because it behaves well with respect to descent. For example, if there are schemes and can be refined to an fppf-cover of , if is flat, locally finite type, or locally of finite presentation, then has this property.[8] this kind of idea can be extended further by considering properties local either on the target or the source of a morphism . For a cover we say a property is local on the source if

has if and only if each has .

There is an analogous notion on the target called local on the target. This means given a cover

has if and only if each has .

For the fppf topology, having an immersion is local on the target.[9] In addition to the previous properties local on the source for the fppf topology, being universally open is also local on the source.[10] Also, being locally Noetherian and Jacobson are local on the source and target 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 is an fpqc-algebraic stack[12]pg 40.

Representable diagonal
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By definition, a 1-morphism of categories fibered in groupoids is representable by algebraic spaces[13] if for any fppf morphism of schemes and any 1-morphism , the associated category fibered in groupoids

is representable as an algebraic space,[14][15] meaning there exists an algebraic space

such that the associated fibered category [16] is equivalent to . There are a number of equivalent conditions for representability of the diagonal[17] which help give intuition for this technical condition, but one of main motivations is the following: for a scheme and objects the sheaf is representable as an algebraic space. In particular, the stabilizer group for any point on the stack 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

the representability of the diagonal is equivalent to being representable for an algebraic space . This is because given morphisms from algebraic spaces, they extend to maps from the diagonal map. There is an analogous statement for algebraic spaces which gives representability of a sheaf on as an algebraic space.[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 -stack for an -stack .

Surjective and smooth atlas

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2-Yoneda lemma
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The existence of an scheme and a 1-morphism of fibered categories which is surjective and smooth depends on defining a smooth and surjective morphisms of fibered categories. Here is the algebraic stack from the representable functor on upgraded to a category fibered in groupoids where the categories only have trivial morphisms. This means the set

is considered as a category, denoted , with objects in as morphisms

and morphisms are the identity morphism. Hence

is 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 .

Representable morphisms of categories fibered in groupoids
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To say this morphism is smooth or surjective, we have to introduce representable morphisms.[20] A morphism of categories fibered in groupoids over is said to be representable if given an object in and an object the 2-fibered product

is representable by a scheme. Then, we can say the morphism of categories fibered in groupoids is smooth and surjective if the associated morphism

of schemes is smooth and surjective.

Deligne-Mumford stacks

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Algebraic stacks, also known as Artin stacks, are by definition equipped with a smooth surjective atlas , where is the stack associated to some scheme . If the atlas is moreover étale, then 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 , the moduli stack of rank 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

sending a scheme to its groupoid of -torsors is representable as a stack over the Etale topology, but the Picard-stack of -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 the Kummer sequence

is exact only as a sequence of fppf sheaves, but not as a sequence of etale sheaves.

Defining algebraic stacks over other topologies

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Using other Grothendieck topologies on 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

of big topologies on .

Structure sheaf

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The structure sheaf of an algebraic stack is an object pulled back from a universal structure sheaf on the site .[21] This universal structure sheaf[22] is defined as

and the associated structure sheaf on a category fibered in groupoids

is defined as

where comes from the map of Grothendieck topologies. In particular, this means is lies over , so , then . As a sanity check, it's worth comparing this to a category fibered in groupoids coming from an -scheme for various topologies.[23] For example, if

is a category fibered in groupoids over , the structure sheaf for an open subscheme gives

so this definition recovers the classic structure sheaf on a scheme. Moreover, for a quotient stack , the structure sheaf this just gives the -invariant sections

for in .[24][25]

Examples

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Classifying stacks

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Many classifying stacks for algebraic groups are algebraic stacks. In fact, for an algebraic group space over a scheme which is flat of finite presentation, the stack is algebraic[4]theorem 6.1.

See also

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References

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  1. ^ A'Campo, Norbert; Ji, Lizhen; Papadopoulos, Athanase (2016-03-07). "On Grothendieck's construction of Teichmüller space". arXiv:1603.02229 [math.GT].
  2. ^ Grothendieck, Alexander; Raynaud, Michele (2004-01-04). "Revêtements étales et groupe fondamental (SGA 1). Expose VI: Catégories fibrées et descente". arXiv:math.AG/0206203.
  3. ^ Giraud, Jean (1971). "II. Les champs". Cohomologie non abélienne. Grundlehren der mathematischen Wissenschaften. Vol. 179. pp. 64–105. doi:10.1007/978-3-662-62103-5. ISBN 978-3-540-05307-1.
  4. ^ a b Artin, M. (1974). "Versal deformations and algebraic stacks". Inventiones Mathematicae. 27 (3): 165–189. Bibcode:1974InMat..27..165A. doi:10.1007/bf01390174. ISSN 0020-9910. S2CID 122887093.
  5. ^ "Section 92.16 (04T3): From an algebraic stack to a presentation—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  6. ^ "Section 34.7 (021L): The fppf topology—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  7. ^ "Section 92.12 (026N): Algebraic stacks—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  8. ^ "Lemma 35.11.8 (06NB)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  9. ^ "Section 35.21 (02YL): Properties of morphisms local in the fppf topology on the target—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  10. ^ "Section 35.25 (036M): Properties of morphisms local in the fppf topology on the source—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  11. ^ "Section 35.13 (034B): Properties of schemes local in the fppf topology—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  12. ^ Goerss, Paul. "Quasi-coherent sheaves on the Moduli Stack of Formal Groups" (PDF). Archived (PDF) from the original on 29 August 2020.
  13. ^ "Section 92.9 (04SX): Morphisms representable by algebraic spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  14. ^ "Section 92.7 (04SU): Split categories fibred in groupoids—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  15. ^ "Section 92.8 (02ZV): Categories fibred in groupoids representable by algebraic spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  16. ^ is the embedding sending a set to the category of objects and only identity morphisms. Then, the Grothendieck construction can be applied to give a category fibered in groupoids
  17. ^ "Lemma 92.10.11 (045G)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  18. ^ "Section 78.5 (046I): Bootstrapping the diagonal—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
  19. ^ Simpson, Carlos (1996-09-17). "Algebraic (geometric) n-stacks". arXiv:alg-geom/9609014.
  20. ^ "Section 92.6 (04ST): Representable morphisms of categories fibred in groupoids—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-03.
  21. ^ "Section 94.3 (06TI): Presheaves—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
  22. ^ "Section 94.6 (06TU): The structure sheaf—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
  23. ^ "Section 94.8 (076N): Representable categories—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
  24. ^ "Lemma 94.13.2 (076S)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
  25. ^ "Section 76.12 (0440): Quasi-coherent sheaves on groupoids—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
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Artin's Axioms

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Papers

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Applications

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Other

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