Ext (homological algebra)
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| Ext (homological algebra) | |
|---|---|
| Name | Ext (homological algebra) |
| Field | Homological algebra |
| Introduced | 1940s |
| Founders | Samuel Eilenberg; Saunders Mac Lane |
Ext (homological algebra) Ext is a family of derived functors measuring extensions of modules and objects in abelian categories. It refines notions from Samuel Eilenberg and Saunders Mac Lane's work in Homological algebra and plays central roles in Category theory, Algebraic topology, and Algebraic geometry through connections with cohomology, derived categories, and classification of extensions.
Definition and basic properties
For an abelian category such as modules over a ring R-mod, Ext^n(A,B) is the right-derived functor of Hom(−,B) or Hom(A,−) depending on variance conventions; this concept originates in the work of Cartan–Eilenberg and the text by Henri Cartan and Samuel Eilenberg. Ext^0(A,B) ≅ Hom(A,B), while Ext^1(A,B) parametrizes equivalence classes of extensions of A by B, relating to constructions in Emmy Noether's structural algebra. Basic properties include long exact sequences arising from short exact sequences, vanishing criteria for projective and injective objects, and dimension-shifting results used in Jean-Pierre Serre's cohomological methods and in computations for Alexander Grothendieck's schemes.
Construction via projective and injective resolutions
Ext can be computed using projective resolutions of the first variable or injective resolutions of the second variable, techniques appearing in Hodge theory-adjacent literature and classical expositions by Hyman Bass and Harold Simmons. Given a projective resolution P_•→A, apply Hom(P_•,B) and take cohomology to obtain Ext^n(A,B); dually, given an injective resolution B→I^•, apply Hom(A,I^•). These constructions are foundational in treatments by Jean-Louis Verdier and underpin the formalism of Derived functors used by Alexander Grothendieck in developing Sheaf cohomology and in the formation of the Derived category by Grothendieck and Bernard Keller.
Universal δ-functor characterization
Ext^n(−,−) is characterized as a universal δ-functor in the sense of Jean-Louis Verdier and Beno Eckmann's collaborators, uniquely determined up to unique isomorphism by its universal property and by the vanishing on injective or projective objects. This universality is explicated in foundational texts by Charles Weibel and in expositions by Peter Hilton and Ulrich Stuhler, linking Ext to Grothendieck's axiomatic approach to derived functors and to uniqueness results used by Michael Atiyah and Isadore Singer in index-theoretic contexts.
Computation and examples
Concrete computations occur in many classical settings: for modules over a principal ideal domain such as Euclid's integers Z, Ext^1(Z/nZ, Z) ≅ Z/nZ, a calculation appearing in textbooks by David Eisenbud and Serge Lang. For group cohomology, Ext relates to H^n(G, M) for a group Évariste Galois-context via projective resolutions of the trivial module, with computations in the work of Noether and Emmy Noether's successors. In algebraic topology, Ext groups appear in the Adams spectral sequence computations for stable homotopy groups of spheres studied by J. Frank Adams and Haynes Miller. Examples include Ext over the Steenrod algebra in Milnor's and Moore's work, and calculations for coherent sheaves on Projective space in treatments by Robin Hartshorne.
Long exact sequences and functoriality
Ext yields long exact sequences in both arguments: a short exact sequence 0→A'→A→A''→0 induces long exact sequences of Ext^n(−,B) and Ext^n(A,−) with connecting homomorphisms δ, a mechanism exploited by Jean Leray and formalized in Samuel Eilenberg-style homological algebra. Functoriality under morphisms in each variable is natural and is used in comparisons of spectral sequences such as those of Leray–Serre and in derived-category morphisms treated by Amnon Neeman. Compatibility with change-of-ring and adjunctions appears in contexts studied by Hyman Bass and Donald Knuth in algebraic algorithmic implementations.
Yoneda interpretation and extensions
Yoneda's viewpoint identifies Ext^1(A,B) with equivalence classes of short exact sequences 0→B→E→A→0, an interpretation due to Nicolai Bourbaki-style formalism and developed in expositions by Claude Chevalley and Saunders Mac Lane. Higher Ext^n correspond to n-fold extensions or chains of extensions, classified via Yoneda composition that composes extension classes, a categorical construction central to Pierre Deligne's work and appearing in Grothendieck's theory of derived functors. This perspective connects with obstruction theory in René Thom-style topology and with deformation theory as treated by Michael Artin.
Applications in algebra and topology
Ext is ubiquitous: it classifies extensions in module theory and group theory, computes obstructions in deformation and obstruction theories used by Alexander Grothendieck and Mikhail Gromov, appears in spectral sequences such as the Adams spectral sequence in stable homotopy theory pioneered by J. Frank Adams, organizes derived-category invariants used by Max Lieblich and Jacob Lurie in derived algebraic geometry, and enters duality theorems like Serre duality and Grothendieck duality central to Algebraic geometry literature. Applications extend to representation theory of Richard Brauer-type algebras, classification problems in Number theory via Galois cohomology of Évariste Galois-related extensions, and categorical approaches exploited in modern research by Tom Bridgeland and Keller.