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| Localization theorem | |
|---|---|
| Name | Localization theorem |
| Field | Mathematics |
| Subfield | Algebraic topology; Algebraic geometry; Homological algebra |
| Introduced | 20th century |
| Notable for | Relating global invariants to fixed-point or localized data |
Localization theorem
The Localization theorem is a collection of results in Mathematics that relate global invariants of a space, scheme, or module to data concentrated on a subspace, fixed points, or a multiplicatively localized ring. It appears in contexts ranging from Algebraic topology and Equivariant cohomology to Algebraic geometry and Homological algebra, and connects constructions in the traditions of Henri Cartan, Jean Leray, Atiyah–Bott, and Grothendieck.
The Localization theorem unifies themes in the works of Henri Cartan, Jean Leray, Michael Atiyah, Raoul Bott, Alexander Grothendieck, and Pierre Deligne by giving techniques to replace a global object by its restriction to a smaller locus such as a fixed-point set under a group action or a closed subscheme. It features in formulations involving equivariant cohomology, K-theory, Borel–Moore homology, and derived categories as in the schools of Samuel Eilenberg, Saunders Mac Lane, and Daniel Quillen.
One classical form is the Localization Theorem in Equivariant cohomology: for a compact Lie group like Circle group or torus action on a compact manifold as in the setting of Atiyah–Bott fixed-point theorem and Berline–Vergne formula, the inclusion of the fixed-point set induces an isomorphism after inverting appropriate Euler classes in the cohomology ring; this variant links to results of Friedrich Hirzebruch and Bott periodicity theorem. In Algebraic geometry a localization sequence in Chow groups or K-theory gives long exact sequences relating the groups of a scheme, a closed subscheme, and its open complement as developed by Grothendieck and applied by William Fulton. In Commutative algebra and Homological algebra there is localization of rings and modules at a multiplicative set, yielding exactness properties exploited by Oscar Zariski and later by Jean-Pierre Serre in the context of Spec and coherent sheaves.
Proof approaches vary: the equivariant topological proofs use spectral sequences and Mayer–Vietoris arguments in the style of Jean Leray and Jean-Pierre Serre, and localization via the Atiyah–Bott technique employs deformation to the normal cone and analysis on tubular neighborhoods akin to methods by Raoul Bott and Michael Atiyah. Algebraic proofs rely on localization of rings and derived functors as in the work of Alexander Grothendieck, using derived categories popularized by Pierre Deligne and techniques from Homological algebra introduced by Samuel Eilenberg and Saunders Mac Lane. K-theoretic proofs were refined by Daniel Quillen using higher algebraic K-theory and by later contributors linked to the Beilinson–Bernstein–Deligne formalism. Analytic approaches for elliptic complexes connect to the analysis of pseudodifferential operators in the tradition of Atiyah–Singer index theorem.
Localization results are pivotal in computing invariants in enumerative geometry as in the Gromov–Witten invariants program and in mirror symmetry contexts influenced by Maxim Kontsevich and Edward Witten. In Symplectic geometry and Hamiltonian dynamics, the localization theorem underpins the Duistermaat–Heckman formula and techniques used by researchers following Victor Guillemin and Shlomo Sternberg. In representation theory and geometric representation theory, localization techniques connect to the Beilinson–Bernstein localization of Lie algebra representations and to character formulae studied by Harish-Chandra and David Kazhdan. Computational uses include effective calculation of characteristic classes exploited by William Fulton and use in fixed-point localization in moduli space problems concerning Donaldson–Thomas theory, Seiberg–Witten invariants, and moduli studied by Simon Donaldson.
- Equivariant cohomology: a circle action on a compact manifold with isolated fixed points yields the Atiyah–Bott localization formula, applied in examples treated by Michael Atiyah and Raoul Bott to compute integrals on projective spaces and flag varieties related to Élie Cartan's work. - Algebraic K-theory: localization sequences for a closed immersion and its complement in a scheme are used in computations by William Fulton and Daniel Quillen for projective varieties and singular schemes encountered in the work of Alexander Grothendieck. - Commutative algebra: localization at a prime ideal of a Noetherian ring simplifies dimension-theoretic problems studied by Oscar Zariski and Pierre Samuel. - Symplectic geometry: Hamiltonian torus actions on compact symplectic manifolds produce Duistermaat–Heckman measures, exemplified in cases analyzed by Victor Guillemin and Hans Duistermaat.
The concept evolved from localization of functions and modules in 19th-century algebra, advanced by figures like Oscar Zariski and André Weil, and was given cohomological form via spectral sequence methods by Jean Leray during the 1940s. The algebraic geometry formalism was consolidated by Alexander Grothendieck in the 1950s and 1960s through his foundational work in the EGA school and subsequent developments by Pierre Deligne. Topological fixed-point localization crystallized in the 1960s and 1970s with the contributions of Michael Atiyah, Raoul Bott, and later refinements by researchers in equivariant cohomology and K-theory including Daniel Quillen and contributors to the Atiyah–Bott fixed-point theorem lineage. Modern applications expanded through the influence of Maxim Kontsevich and Edward Witten in mathematical physics and enumerative geometry.
Category:Algebraic topology Category:Algebraic geometry Category:Homological algebra