Algebraic stack

In algebraic geometry, an algebraic stack is a concept introduced to generalize algebraic varieties, schemes, and algebraic spaces. They were originally proposed in a 1969 paper^[1] by Pierre Deligne and David Mumford to define the (fine) moduli space of genus g curves; their definition is currently referred to as Deligne–Mumford stacks. When viewed in this light, algebraic stacks are an algebraic analogue of orbifolds. They were generalized by Michael Artin in 1974^[2] to what is now called an Artin stack, or sometimes, confusingly, an algebraic stack.

1 Motivation
2 Formal definitions
- 2.1 Deligne–Mumford stacks
- 2.2 Artin stacks
3 Properties
4 Examples
5 See also
6 Notes
7 References
8 External links

Motivation

When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient. For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes.

In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. Constructions of moduli spaces often proceed by first constructing a larger space parametrizing the objects in question, and then quotienting by a group action to account for objects with automorphisms which have been overcounted.

Formal definitions

A stack is a category X over the étale site satisfying the following three properties.

We can define restrictions of objects over a scheme S to objects in open coverings of S: The category X is fibered in groupoids over the étale site.
We can patch isomorphisms: Isomorphisms are a sheaf for X.
We can patch objects: Every descent datum is effective.

Note that the étale site is the name for the usual category of schemes considered together with the étale Grothendieck topology.

Technically an algebraic stack is a stack that can be suitably "covered" by algebraic spaces with respect to an appropriate Grothendieck topology.

Deligne–Mumford stacks

A stack, as defined above, is a Deligne–Mumford stack if there is an étale and surjective representable morphism from (the stack associated to) a scheme to X. A morphism Y $\rightarrow$ X of stacks is representable if, for every morphism S $\rightarrow$ X from (the stack associated to) a scheme to X, the fiber product Y ×_X S is isomorphic to (the stack associated to) a scheme. The fiber product of stacks is defined using the usual universal property, and changing the requirement that diagrams commute to the requirement that they 2-commute.

Deligne–Mumford stacks can be thought of as restricting the stabilizer groups of points to be finite.

Artin stacks

A stack, as defined above, is an Artin stack if there exists a smooth and surjective representable morphism from (the stack associated to) a scheme to X.

Artin stacks can be thought of as restricting the stabilizer groups to be algebraic groups.

Properties

More generally a stack refers to any category acting more or less like a moduli space with a universal family (analogous to a classifying space) parametrizing a family of related mathematical objects such as schemes or topological spaces, especially when the members of these families have nontrivial automorphisms. This leads to the notion that the points of the stack should carry automorphisms themselves, and this in turn gives rise to the notion of a stack as a certain kind of "category fibered in groupoids".

Moduli spaces which do not carry this extra information are then referred to as coarse moduli spaces and stacks then act as relatively fine moduli spaces.

Examples

The moduli space of algebraic curves (Deligne–Mumford stack) defined as a universal family of curves of given genus g does not exist as an algebraic variety because in particular there are elliptic curves admitting nontrivial automorphisms. For elliptic curves over the complex numbers the corresponding stack is a quotient of the upper half-plane by the action of the modular group.

Notes

^ Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS 36: 75–109.
^ Artin, Michael (1974), "Versal deformations and algebraic stacks", Inventiones Mathematicae 27 (3): 165–189, doi:10.1007/BF01390174.

References

Gómez, Tomás L. (2001), "Algebraic stacks", Indian Academy of Sciences. Proceedings. Mathematical Sciences 111 (1): 1–31, arXiv:math/9911199, doi:10.1007/BF02829538, MR 1818418
Laszlo, Yves; Olsson, Martin (2008), "The six operations for sheaves on Artin stacks. I. Finite coefficients", Institut des Hautes Études Scientifiques. Publications Mathématiques 107 (107): 109–168, doi:10.1007/s10240-008-0011-6, MR 2434692
Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 39, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65761-3, MR 1771927 Unfortunately this book uses the incorrect assertion that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. Some of these errors were fixed by Olsson (2007).
Olsson, Martin (2007), "Sheaves on Artin stacks", Journal für die Reine und Angewandte Mathematik. [Crelle's Journal] 603 (603): 55–112, doi:10.1515/CRELLE.2007.012, MR 2312554