Stack (mathematics)
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| Stack (mathematics) | |
|---|---|
| Name | Stack (mathematics) |
| Field | Algebraic geometry; Category theory |
| Introduced by | Alexandre Grothendieck; Jean Giraud |
| Year | 1960s–1970s |
Stack (mathematics)
A stack in mathematics is a categorical gadget blending ideas from Category theory and Algebraic geometry to encode families of objects with automorphisms across varying bases. Originating in work of Alexandre Grothendieck and formalized by Jean Giraud and others, stacks generalize sheaves, fibered categories, and groupoids to treat moduli problems that fail to be represented by schemes or algebraic spaces. Stacks appear throughout modern work related to Mumford, Deligne, Artin, and interact with concepts from Grothendieck topology, Étale cohomology, and Derived category techniques.
Introduction
A stack is defined with respect to a Grothendieck topology (for example the Zariski topology, Étale topology, or fppf topology) on a base category such as the category of schemes or complex analytic spaces. It enhances the notion of a sheaf by allowing values in groupoids rather than sets, thereby tracking automorphisms that occur in families like vector bundles, principal bundles, or coherent sheaves. Stacks are central to modern formulations of moduli theory used by researchers influenced by Mumford's geometric invariant theory, Deligne's work on curves, and Artin's algebraic criteria.
Definitions and Examples
Formally, a stack is a fibered category satisfying effective descent for objects and isomorphisms with respect to a chosen topology. Classical examples include the stack of vector bundles on a fixed scheme (related to Grothendieck's original visions), the Picard stack connected to Jacobian variety constructions, and the moduli stack of curves central to work by Deligne and Mumford. Other prominent examples are the classifying stack BG for a linear algebraic group G studied in relation to Chevalley and Weyl theory, and the stack of principal G-bundles appearing in work by Atiyah and Bott on gauge theory. Analytic analogues relate to the moduli of complex structures and the Teichmüller theory pursued by researchers following Ahlfors and Bers.
Fibered Categories and Descent
Stacks are built on the language of fibered categories over a base category of spaces; the fiber over a scheme parametrizes objects on that scheme. The descent condition connects to classical results such as the descent theory of Grothendieck and the faithfully flat descent developed by authors like Serre and Raynaud. Effective descent for objects and morphisms ensures that families glued from local data yield global objects, paralleling constructions in work by Steinberg and techniques used in the proof of the representability theorems by Artin. The role of groupoids and 2-categorical coherence is essential, echoing formal structures studied by Eilenberg and Mac Lane.
Algebraic Stacks and Deligne–Mumford Stacks
An algebraic stack (often called an Artin stack) satisfies additional conditions rendering it amenable to algebraic-geometric techniques: diagonals that are representable and existence of smooth or syntomic atlases from schemes or algebraic spaces. Deligne–Mumford stacks impose further finiteness on automorphism groups, producing an étale atlas and enabling the use of techniques from the theories developed by Deligne and Mumford for moduli of curves. Key constructions include the moduli stack of stable curves used in the proof of the Mumford conjectures, and quotient stacks arising from group actions studied in contexts related to Geometric Invariant Theory and work of Mumford, Kirwan, and Thaddeus.
Morphisms, Properties, and Representability
Morphisms of stacks generalize morphisms of schemes and require careful handling of 2-morphisms and fibered structures. Properties such as representability by schemes or algebraic spaces, being separated, proper, smooth, or étale are formulated via diagonal maps and atlases, in analogy with criteria established by Grothendieck and refined by Artin and Keel. Representability criteria often employ deformation and obstruction theories developed by Schlessinger and formal methods linked to Illusie's cotangent complex. Important invariants include stabilizer groups (in line with work by Serre on group schemes) and inertia stacks used in Riemann–Roch type formulas and orbifold analyses pioneered in part by Thomason and Abramovich.
Cohomology and Intersection Theory on Stacks
Cohomological tools for stacks extend Étale cohomology and singular cohomology via techniques adapted from the theories of Grothendieck, Deligne, and Verdier. Intersection theory on stacks parallels Fulton’s intersection theory on schemes and incorporates contributions from stabilizers; such approaches have been developed by Fulton, Behrend, and Vistoli in constructing virtual fundamental classes and Gromov–Witten invariants. Derived and motivic enhancements connect stacks to ideas from Kontsevich and Toen in derived algebraic geometry, while Riemann–Roch theorems for stacks build on foundations laid by Grothendieck and later extended by Edidin and Graham.
Applications and Examples in Moduli Theory
Stacks provide the natural language for many moduli problems: the moduli of curves, vector bundles, coherent sheaves, and stable maps central to enumerative geometry and string-theoretic enumerations developed by Witten and Kontsevich. The construction of moduli of polarized varieties and K-stability criteria interacts with contributions from Tian and Donaldson. Quotient stacks model orbifolds and global quotient constructions related to Satake and Chen–Ruan orbifold cohomology, while stacks also underpin arithmetic moduli considerations as in the work of Faltings and Mazur.