Algebraic stack
In mathematics, an algebraic stack is a vast generalization of
Definition
Motivation
One of the motivating examples of an algebraic stack is to consider a
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
Algebraic stacks
It turns out using the
such that
- is a category fibered in groupoids, meaning the overcategoryfor 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.
Explanation of technical conditions
Using the fppf topology
First of all, the fppf-topology is used because it behaves well with respect to
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
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
2-Yoneda lemma
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
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
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
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
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
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
Examples
Classifying stacks
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
- Gerbe
- Chow group of a stack
- Cohomology of a stack
- Quotient stack
- Sheaf on an algebraic stack
- Toric stack
- Artin's criterion
- Pursuing Stacks
- Derived algebraic geometry
References
- arXiv:1603.02229 [math.GT].
- arXiv:math.AG/0206203.
- ISBN 978-3-540-05307-1.
- ^ S2CID 122887093.
- ^ "Section 92.16 (04T3): From an algebraic stack to a presentation—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ "Section 34.7 (021L): The fppf topology—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ "Section 92.12 (026N): Algebraic stacks—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ "Lemma 35.11.8 (06NB)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ "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.
- ^ "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.
- ^ "Section 35.13 (034B): Properties of schemes local in the fppf topology—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ Goerss, Paul. "Quasi-coherent sheaves on the Moduli Stack of Formal Groups" (PDF). Archived (PDF) from the original on 29 August 2020.
- ^ "Section 92.9 (04SX): Morphisms representable by algebraic spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ "Section 92.7 (04SU): Split categories fibred in groupoids—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ "Section 92.8 (02ZV): Categories fibred in groupoids representable by algebraic spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ 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
- ^ "Lemma 92.10.11 (045G)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- ^ "Section 78.5 (046I): Bootstrapping the diagonal—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29.
- arXiv:alg-geom/9609014.
- ^ "Section 92.6 (04ST): Representable morphisms of categories fibred in groupoids—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-03.
- ^ "Section 94.3 (06TI): Presheaves—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
- ^ "Section 94.6 (06TU): The structure sheaf—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
- ^ "Section 94.8 (076N): Representable categories—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
- ^ "Lemma 94.13.2 (076S)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
- ^ "Section 76.12 (0440): Quasi-coherent sheaves on groupoids—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-01.
External links
Artin's Axioms
- https://stacks.math.columbia.edu/tag/07SZ - Look at "Axioms" and "Algebraic stacks"
- Artin Algebraization and Quotient Stacks - Jarod Alper
Papers
- Alper, Jarod (2009). "A Guide to the Literature on Algebraic Stacks" (PDF). S2CID 51803452. Archived from the original(PDF) on 2020-02-13.
- Hall, Jack; Rydh, David (2014). "The Hilbert stack". S2CID 55936583.
- Behrend, Kai A. (2003). "Derived ℓ-Adic Categories for Algebraic Stacks" (PDF). Memoirs of the American Mathematical Society. 163 (774): 1–93. ISBN 978-1-4704-0372-0.
Applications
- Lafforgue, Vincent (2014). "Introduction to chtoucas for reductive groups and to the global Langlands parameterization". ].
- Deligne, P.; Rapoport, M. (1973). "Les Schémas de Modules de Courbes Elliptiques". Modular Functions of One Variable II. Lecture Notes in Mathematics. Vol. 349. pp. 143–316. ISBN 978-3-540-06558-6.
- Knudsen, Finn F. (1983). "The projectivity of the moduli space of stable curves, II: The stacks ". Mathematica Scandinavica. 52: 161. .
- Jiang, Yunfeng (2019). "On the construction of moduli stack of projective Higgs bundles over surfaces". arXiv:1911.00250 [math.AG].