Category of chain complexes
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| Category of chain complexes | |
|---|---|
| Name | Category of chain complexes |
| Field | Homological algebra |
| Introduced | 20th century |
| Notable | Derived category, Homotopy category, Differential graded category |
Category of chain complexes
The category of chain complexes is a foundational construction in homological algebra linking Alexander Grothendieck, Henri Cartan, Samuel Eilenberg, Saunders Mac Lane, and Jean-Louis Verdier to modern tools such as the derived category, triangulated category, differential graded algebra, spectral sequence, and homological functor. It organizes chain complex objects and chain map morphisms to study invariants like homology groups, connecting to results of Emmy Noether, David Hilbert, Henri Poincaré, Élie Cartan, and techniques used in the work of Jean-Pierre Serre and Alexander Grothendieck.
Definition and basic properties
A chain complex in an abelian category such as Mod-R or Abelian group is a sequence of objects and differentials satisfying d∘d=0; the category of such complexes, often denoted Ch(A), is itself an additive category, admits kernels and cokernels, and inherits exact sequence notions related to short exact sequence and long exact sequence constructions used by Henri Cartan, Samuel Eilenberg, and Saunders Mac Lane. Fundamental properties include the existence of degreewise finite limits and colimits, compatibility with projective and injective resolutions as in the work of Cartan-Eilenberg, and interactions with tensor product and Hom functor structures central to Grothendieck's derived functor formalism and Jean-Louis Verdier's triangulated frameworks.
Morphisms, homotopy, and quasi-isomorphisms
Morphisms are degree-preserving chain maps; two morphisms may be related by a chain homotopy, paralleling homotopy theories developed by J. H. C. Whitehead and Daniel Quillen, while quasi-isomorphisms—maps inducing isomorphisms on homology groups—play the role of weak equivalences in model structures studied by Quillen, Mark Hovey, and Jacob Lurie. The interplay of chain homotopy, mapping cylinder constructions, projective resolutions, and the localization at quasi-isomorphisms underlies equivalences used by Jean-Pierre Serre, Alexander Grothendieck, and Bernhard Keller in derived and dg-contexts.
Constructions: shifts, cones, and mapping cones
The shift (or suspension) functor generalizes suspension maps from Algebraic topology and appears in homotopical algebra work of P. S. Alexandroff and J. H. C. Whitehead; cones and mapping cones provide explicit triangles similar to constructions in Verdier's triangulated categories and are central in the treatments by Cartan-Eilenberg and Gelfand–Manin. Mapping cone constructions relate to distinguished triangles studied by Jean-Louis Verdier, Robin Hartshorne, and Bernhard Keller, and are used in proofs appearing in texts by Weibel, Gelfand, and Manin.
Derived category and homotopy category
Localizing the category of chain complexes at quasi-isomorphisms yields the derived category, a notion introduced by Grothendieck and formalized by Verdier; the homotopy category arises by passing to chain homotopy classes and links to the work of Heller and Hovey on model categories. The derived category underpins coherent duality theorems of Grothendieck and Hartshorne, interacts with Brown representability related to Adams spectral sequence arguments used by Frank Adams, and is central to modern approaches by Jacob Lurie and Bernhard Keller in higher and dg-enhanced contexts.
Monoidal and abelian structures
When the underlying category is symmetric monoidal such as Mod-R or categories of sheafs over schemes studied by Grothendieck and Hartshorne, the category of chain complexes inherits a monoidal structure compatible with total complexes, Künneth formulas associated to Emmy Noether ideas, and monoidal model structures examined by Hovey and Jeff Smith. If the base is abelian then Ch(A) is abelian with exactness conditions reflecting the abelian heritage of Noether, Eilenberg, and Mac Lane and with tensor-hom adjunctions important for derived tensor product constructions used by Verdier and Grothendieck.
Examples and applications
Examples include complexes of R-modules in commutative algebra as in work of Emmy Noether, complexes of sheafs on schemes in algebraic geometry following Grothendieck and Hartshorne, and singular chain complexes in algebraic topology associated to Henri Poincaré and Eilenberg–Steenrod axioms. Applications span computations of Ext and Tor in homological algebra texts by Weibel, duality theorems of Grothendieck and Hartshorne, and modern developments in mirror symmetry and Kontsevich's homological mirror symmetry, with dg-enhancements used by Keller and Toen.
Variations and generalizations (dg-categories, filtered complexes)
Generalizations include differential graded categorys studied by Bernhard Keller and Maxim Kontsevich, filtered complexes and spectral sequences examined by Jean Leray and Jean-Pierre Serre, and mixed Hodge complexes developed by Pierre Deligne and applied in perverse sheaf contexts by MacPherson and Beilinson. These variations connect to higher-categorical formalisms of Jacob Lurie, model category approaches of Quillen and Hovey, and enhancements used in derived algebraic geometry by Toën and Vezzosi.