Algebraic stack
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| Algebraic stack | |
|---|---|
| Name | Algebraic stack |
| Field | Algebraic geometry |
| Introduced | 1960s–1970s |
| Notable | Pierre Deligne, Jean-Michel Bismut, Alexander Grothendieck |
Algebraic stack. An algebraic stack is a categorical generalization of schemes and algebraic spaces used to parametrize families with automorphisms, arising in work of Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, and David Mumford. It refines notions developed in the context of the Grothendieck topology, Étale cohomology, and the theory of moduli spaces, connecting to constructions by Michael Artin and formalized through ideas related to the Stacks Project. Algebraic stacks play central roles in the study of moduli of curves, vector bundle classification, and modern intersections of string theory, number theory, and representation theory.
Definition and basic properties
An algebraic stack is defined as a stack over the category of schemes with respect to a Grothendieck topology (often the fppf topology or étale topology) satisfying effective descent and possessing a representable diagonal and a smooth or locally representable cover by a scheme. Foundational criteria were established in work by Michael Artin, with precursors in lectures by Alexander Grothendieck and elaborations by Pierre Deligne and David Mumford. The diagonal condition connects to representability results used in Élie Cartan-style deformation theory and to concepts from Serre duality and Kodaira vanishing in specialized contexts. Algebraic stacks admit notions of stabilizer groups, inertia stacks, and residual gerbes, which relate to constructions by Jean-Michel Bismut in index-theoretic interpretations and to groupoid presentations considered by André Weil and Henri Cartan.
Stacks are 2-functors with descent data encoded in a 2-categorical framework influenced by categorical work of Saunders Mac Lane and Samuel Eilenberg. The representability of the diagonal often requires checking conditions like being quasi-compact, separated, or locally of finite presentation, echoing criteria from the theory of Noetherian schemes developed by Oscar Zariski and André Weil. Many existence proofs use techniques introduced by Grothendieck in the context of the Hilbert scheme and the Picard scheme.
Examples
Classical examples include the moduli stack of algebraic curves introduced by David Mumford and refined by Deligne and Mumford as the stack of stable curves, which is closely linked to the Deligne–Mumford compactification. The stack of vector bundles on a fixed curve appears in the work of Michael Atiyah and Raoul Bott in topological settings and in algebraic guise in studies by Vladimir Drinfeld and Alexander Beilinson. Quotient stacks [X/G] arise from actions of algebraic groups such as GL_n or SL_n on schemes and are central to geometric invariant theory developed by David Mumford and Frances Kirwan. The classifying stack BG for a linear algebraic group G provides examples connected to cohomology theories explored by Jean-Louis Verdier and Pierre Deligne.
Other important instances include the stack of principal G-bundles on a curve studied in relation to the Langlands program by Robert Langlands and Edward Frenkel, the derived enhancement appearing in derived algebraic geometry advanced by Jacob Lurie and Bertrand Toën, and the formalism of Artin stacks used in enumerative geometry related to the Gromov–Witten invariants studied by Kontsevich and Yuri Manin.
Morphisms and 2-categorical structure
Morphisms between algebraic stacks are 1-morphisms in a 2-category, with 2-morphisms encoding natural transformations; this 2-categorical viewpoint has roots in categorical studies by Mac Lane and Jean Bénabou. Properties of morphisms—representable, smooth, étale, unramified, proper, separated, finite presentation—generalize classical notions for morphisms of schemes introduced by Alexander Grothendieck and analyzed in the work of Oscar Zariski and Jean-Pierre Serre. Fibered categories and groupoid presentations by schemes link to groupoid schemes studied by Claude Chevalley and to descent theory that traces back to Grothendieck.
The notion of 2-fiber product and 2-pullback is essential in gluing constructions and base change arguments used in the proof of representability theorems by Michael Artin and in degeneration techniques used by Deligne and Illusie. Stabilizer group schemes and inertia stacks determine stacky behavior under morphisms, with connections to group cohomology developed by Jean Leray and Claude Shannon—in algebraic settings often mediated by methods from Serre and Jean-Pierre Serre.
Geometric properties and conditions
One studies algebraic stacks via geometric conditions: being Deligne–Mumford, Artin (algebraic), smooth, proper, or separated. Deligne–Mumford stacks, introduced by Pierre Deligne and David Mumford, have unramified diagonals and étale covers by schemes; Artin stacks allow smooth covers and were axiomatized by Michael Artin. Properness criteria for stacks extend the valuative criterion originally formulated for schemes by Alexander Grothendieck and used in compactification results like the Deligne–Mumford compactification. Cohomological finiteness, boundedness, and local structure theorems rely on techniques developed by Hartshorne and Grothendieck and often invoke deformation-theoretic methods pioneered by Kodaira and Spencer.
Geometric invariant theory by David Mumford informs stability conditions for stacky moduli, while intersection-theoretic invariants such as virtual fundamental classes connect to the enumerative frameworks of Maxim Kontsevich and Richard Thomas. The study of gerbes over schemes and twisted sheaves is influenced by work of Jean Giraud and Alexander Grothendieck on non-abelian cohomology.
Moduli applications and constructions
Algebraic stacks provide the natural language for moduli problems with automorphisms: moduli of curves (Deligne–Mumford stack), vector bundles (Moduli of vector bundles), principal bundles (central in the geometric Langlands program), and stable maps used in Gromov–Witten theory by Kontsevich and Behrend. Constructions often proceed via representability theorems of Artin and involve obstruction theories developed by Illusie and Michael Thaddeus. Stacks arise in arithmetic contexts such as the study of Shimura varieties and integral models investigated by Pierre Deligne and George Pappas, and in derived settings through work of Jacob Lurie, Bertrand Toën, and Gabriele Vezzosi.
Compactifications, stabilization, and wall-crossing phenomena relate to techniques by Frances Kirwan and Dolgachev–Hu notions adapted to stacky contexts, while categorical viewpoints informed by Maxim Kontsevich and Dmitri Orlov link stacks to homological mirror symmetry and derived categories. Moduli stacks underpin modern interactions between algebraic geometry and mathematical physics including string dualities studied by Edward Witten and enumerative predictions from Mirror Symmetry.