homological algebra
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| homological algebra | |
|---|---|
| Name | Homological algebra |
| Field | Mathematics |
| Introduced | 1940s |
| Notable people | * Samuel Eilenberg * Saunders Mac Lane * Jean-Pierre Serre * Henri Cartan * Claude Chevalley * Alexander Grothendieck * Henri Cartan * Daniel Quillen * Pierre Deligne * Spencer Bloch * Jean-Louis Verdier |
homological algebra Homological algebra is a branch of mathematics that studies algebraic structures via sequences of algebraic objects and maps, extracting invariants from exactness properties. It originated in the mid‑20th century through work by Samuel Eilenberg, Saunders Mac Lane, Henri Cartan, and Jean-Pierre Serre, and was expanded by contributions from Alexander Grothendieck, Daniel Quillen, and Jean-Louis Verdier. The subject underpins modern developments in Algebraic topology, Algebraic geometry, Representation theory, and Category theory, providing tools like derived functors and triangulated categories.
Introduction
Homological algebra arose from problems in Algebraic topology such as computing homology groups for spaces related to the Poincaré conjecture and studying cohomology operations appearing in work of Henri Cartan and Jean-Pierre Serre. Early formalism was established by Samuel Eilenberg and Saunders Mac Lane in their treatment of homological properties of functors and natural transformations, influencing later foundations by Alexander Grothendieck in Algebraic geometry and the formulation of spectral sequences used by Jean Leray and Jean-Louis Koszul.
Chain Complexes and Homology
Chain complexes—sequences of modules or objects connected by boundary maps—are central, as developed in contexts like Singular homology and Simplicial homology studied by Poincaré and formalized by Eilenberg and Mac Lane. Homology functors assign graded objects (homology groups) to chain complexes, yielding invariants that were used in the classification problems tackled by Emmy Noether and further applied in computations by Henri Poincaré and Lefschetz in fixed point theorems. Exact sequences, long exact sequences, and mapping cones provide algebraic mechanisms mirroring geometric constructions explored by Lefschetz and later by Grothendieck in cohomological frameworks.
Derived Functors and Ext/Tor
Derived functors formalize how non‑exact functors produce higher cohomological information; canonical examples are Ext and Tor, which trace to projective and injective resolutions developed by Cartan and Eilenberg. Tools such as spectral sequences—originating in work of Jean Leray and refined by Henri Cartan and Jean-Pierre Serre—enable computation of derived functors in complex situations encountered in Algebraic geometry problems addressed by Grothendieck and in algebraic problems studied by Noether. The interplay of Ext and Tor with cup and cap products was integral in advances by Edwin Spanier and influenced later categorical treatments by Mac Lane and Quillen.
Homological Dimensions and Resolutions
Homological dimensions (projective, injective, flat) quantify how far modules or objects deviate from having simple resolutions, topics developed through work of Emmy Noether, David Hilbert, and Cartan. Resolutions—projective, injective, and flat—are used to compute derived functors and were systematized in Grothendieck’s seminars, influencing studies in Commutative algebra by Oscar Zariski and André Weil. Concepts such as global dimension and homological conjectures were focal points in research by Jean-Pierre Serre and later results by Auslander and Bridger in Representation theory and module theory.
Triangulated and Derived Categories
The abstraction of derived categories and triangulated structures was introduced by Jean-Louis Verdier in the setting of Derived category theory, following categorical foundations laid by Grothendieck and Mac Lane. Triangulated categories encode the homological behavior of mapping cones and exact triangles, forming the language for modern treatments in Algebraic geometry (e.g., in work by Pierre Deligne), Representation theory (as in studies by Bernhard Keller), and Topological K‑theory influenced by Daniel Quillen. Enhancements to differential graded categories and model categories trace to constructions by Quillen and later refinements by Kontsevich in contexts such as homological mirror symmetry.
Applications and Connections to Other Fields
Homological algebra techniques permeate multiple domains: in Algebraic topology for spectral sequences and stable homotopy theory used by Adams; in Algebraic geometry for cohomology theories and Grothendieck duality studied by Hartshorne and Deligne; in Representation theory for cohomology of algebras and Auslander‑Reiten theory examined by Maurice Auslander; in Number theory via étale cohomology employed by Grothendieck and Jean-Pierre Serre; and in mathematical physics through derived categories appearing in work of Maxim Kontsevich on homological mirror symmetry. Contemporary research continues in interactions with Noncommutative geometry as in work influenced by Alain Connes and structural developments in higher category theory prompted by Jacob Lurie.