Algebraic stack
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In algebraic geometry, a branch of mathematics, an algebraic stack is a concept introduced to generalize algebraic varieties, schemes, and algebraic spaces. They were originally proposed in Deligne-Mumford (1969) 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 Artin (1974) to what is now called an Artin stack, or sometimes, confusingly, an algebraic stack.
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[edit] Motivation
To be expanded:
- Quotients with non-trivial stabilisers of points
- Moduli spaces of objects with automorphisms
[edit] 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.
To add: features common to both variants.
Technically an algebraic stack is a stack that can be suitably "covered" by algebraic spaces with respect to an appropriate Grothendieck topology.
[edit] 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 X to Y of stacks is representable if, for every morphism S to Y from (the stack associated to) a scheme to Y, the fiber product X cross_Y 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 dimensional.
[edit] 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.
[edit] 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.
[edit] 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 geometrical factor of the upper half-plane by the action of the modular group.
[edit] References
- 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: 165–189.
[edit] External links
- Stacks for everybody
- What is... a Stack?
- In-progress book of Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch
- Web-based algebraic stacks project
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