Short exact sequence

Short exact sequence
Leonardo Almeida Lessa · CC BY-SA 4.0 · source
Name	Short exact sequence
Field	Homological algebra
Introduced	1940s
Notable	Auslander, Grothendieck, Serre

Short exact sequence

A short exact sequence is a concise arrangement in algebraic contexts describing how objects and morphisms from categories like Abelian categories, category theory and homological algebra fit together via kernel and image conditions. It encapsulates structural relations used across mathematics, appearing in work by Alexander Grothendieck, Jean-Pierre Serre, André Weil and influencing developments in Algebraic geometry, Algebraic topology, Representation theory and Algebraic K-theory. Short exact sequences are central to constructions in derived categories, spectral sequences and computations of Ext and Tor.

Definition and basic properties

A short exact sequence is a triple of morphisms 0 → A → B → C → 0 in an Abelian category such that the first map is a kernel of the second and the second map is a cokernel of the first, mirroring conditions used in work by Emmy Noether, Claude Chevalley, Emil Artin and Israel Gelfand. Basic properties include that monomorphisms and epimorphisms interact to give five lemma and snake lemma conclusions used in contexts related to Lefschetz fixed-point theorem, Atiyah–Singer index theorem and constructions appearing in Grothendieck-Riemann-Roch theorem studies. Functoriality under additive functors like those considered by Henri Cartan and Samuel Eilenberg preserves exactness; however, left exact and right exact distinctions, studied by Max Karoubi and Daniel Quillen, govern which parts of exactness remain under application of functors.

Examples

Classic examples arise in Group theory with sequences such as 0 → ℤ → ℚ → ℚ/ℤ → 0 used in analyses related to Fermat's Last Theorem antecedents and in Galois theory contexts tied to Évariste Galois-inspired structure. In Module theory over a ring R, sequences 0 → I → R → R/I → 0 for ideals I connect to themes in Noetherian rings and concepts explored by Oscar Zariski and Pierre Samuel. Vector space examples 0 → U → V → V/U → 0 appear in Hermann Weyl and Élie Cartan-inspired representation problems, while sequences in Sheaf theory such as 0 → O_X(−D) → O_X → O_D → 0 are central in Alexander Grothendieck's work on schemes and in applications to the Riemann–Roch theorem and studies by Jean-Pierre Serre.

Splitting and split exact sequences

A short exact sequence is split if there exists a section or retraction yielding an isomorphism B ≅ A ⊕ C; split conditions are pivotal in classification problems discussed by Nathan Jacobson and in decomposition theorems related to Maschke's theorem in representation theory. Splitness can be characterized by existence of a left inverse or right inverse and features in structural results in Module theory and Category theory frameworks developed by Saunders Mac Lane and Samuel Eilenberg. Non-split sequences underlie obstruction theories and extensions studied by Herman Weyl and later formalized by Igor Dolgachev and Pierre Deligne in geometric contexts.

Exact sequences in homological algebra and derived functors

Short exact sequences generate long exact sequences in homology and cohomology via connecting homomorphisms used in proofs of the Universal Coefficient Theorem, in constructions of Ext and Tor groups, and in spectral sequence arguments of Jean Leray and Gaston Darboux. Derived functors such as those developed by Alexander Grothendieck produce delta maps connecting homological invariants across sequences; these are instrumental in computations appearing in Algebraic topology problems examined by Henri Poincaré and Hassler Whitney. The interplay between short exact sequences and derived categories is central to modern efforts by Amnon Neeman, Maxim Kontsevich and Paul Balmer in triangulated category frameworks and in deriving long exact sequences of Ext and Tor used in computations in Algebraic K-theory by Daniel Quillen.

Short exact sequences of modules and groups

In the categories of R-modules and groups, short exact sequences 0 → N → G → Q → 0 express normal subgroup or submodule embeddings and quotients studied by Niels Henrik Abel antecedents and formalized in group extension theory by Otto Schreier and Hugo Steinhaus. Module-theoretic properties like projectivity and injectivity, investigated by Emmy Noether and Richard Brauer, determine when sequences split; projective modules yield split surjections while injective modules yield split injections. Applications include classification of Finite group extensions relevant to William Burnside and cohomological descriptions via group cohomology developed by Claude Chevalley and Samuel Eilenberg.

Extensions and classification of short exact sequences

Extensions classified by Ext^1(C,A) originate in the work of Samuel Eilenberg, Saunders Mac Lane and Alexander Grothendieck, providing a cohomological parameter space for equivalence classes of extensions 0 → A → B → C → 0. These classification results interlink with obstruction theory in Algebraic topology and deformation theory in Algebraic geometry as pursued by Mikhail Gromov and Maxim Kontsevich. The Baer sum operation on extensions and relationships with Yoneda composition are foundational in studies by Jean-Louis Verdier and facilitate bridge-building between concrete constructions in Module theory and abstract phenomena in derived categories.

Category:Homological algebra