Ext functor
Generated by GPT-5-miniExpansion Funnel Raw 41 → Dedup 0 → NER 0 → Enqueued 0
| Ext functor | |
|---|---|
| Name | Ext functor |
| Field | Homological algebra |
| Introduced | 1940s |
| Authors | Cartan, Eilenberg |
Ext functor The Ext functor is a family of derived functors measuring extension classes of modules and sheaves; it appears in homological algebra, algebraic topology, algebraic geometry, and representation theory. Introduced in the work of Henri Cartan and Samuel Eilenberg, Ext connects to cohomology theories used by figures such as Jean-Pierre Serre, Alexander Grothendieck, and David Hilbert. It serves as a bridge between classical constructions like the Hom functor and modern tools such as the Derived category and Spectral sequence.
Definition and basic properties
For an abelian category with enough projectives or injectives, Ext^n(A,B) is defined as the right derived functors of Hom(−,−) or left derived functors in dual contexts; foundational treatments appear in texts by Henri Cartan, Samuel Eilenberg, Claude Chevalley, and Emmy Noether. The groups Ext^0(A,B) ≅ Hom(A,B) and Ext^n vanish for n<0; they satisfy long exact sequences associated to short exact sequences studied by Emily Noether-era algebraists and used by Emmy Noether’s contemporaries. Key properties include functoriality in both arguments, Baer sum composition rules attributed to Reinhold Baer, and dimension-shifting techniques employed in work by Jean Leray and Hermann Weyl.
Computation and long exact sequences
Ext groups are computed via projective resolutions (as in David Hilbert’s syzygy concepts) or injective resolutions (used in Grothendieck’s apparatus), and by using Cartan–Eilenberg resolutions developed by Cartan and Eilenberg. Derived-category approaches by Alexander Grothendieck, Jean-Louis Verdier, and Grothendieck’s school allow computation with quasi-isomorphisms and total complexes; spectral sequence tools of Jean Leray, Edwin Spanier, and Jean-Pierre Serre produce five-term exact sequences and filtration arguments. The horseshoe lemma and comparison theorems credited to Cartan and Eilenberg facilitate splice constructions that yield the long exact Ext sequence associated to a short exact sequence of modules, much like boundary maps in the Mayer–Vietoris sequence used by Hermann Weyl and Marston Morse.
Interpretations (extensions, derived functors)
Ext^1(A,B) classifies equivalence classes of extensions 0→B→E→A→0 via Baer’s description linked to Reinhold Baer’s work, while higher Ext groups correspond to n-fold extensions studied by Samuel Eilenberg and Saunders Mac Lane. From the derived-functor viewpoint of Grothendieck and Verdier, Ext arises as Hom in the Derived category: Ext^n(A,B) ≅ Hom_{D}(A,B[n]), a perspective exploited by Pierre Deligne and Alexander Grothendieck in the development of Grothendieck duality. Connections with obstruction theory are evident in applications by John Milnor and J. H. C. Whitehead.
Examples and calculations
Classic calculations include Ext groups over principal ideal domains as in work by Emmy Noether and David Hilbert: for finitely generated modules over a PID, Ext reduces to torsion module computations linked to the Fundamental theorem of finitely generated abelian groups noted by Leopold Kronecker. Over group algebras, Ext computes group cohomology H^n(G,−) studied by Eilenberg and Samuel Eilenberg with extensive contributions from John Tate and Jean-Pierre Serre. In algebraic geometry, Ext for coherent sheaves on projective varieties is computed via resolutions from Alexander Grothendieck’s coherent cohomology theory and examples in the work of Grothendieck, Jean-Pierre Serre, and Robin Hartshorne; Serre duality links Ext^{dim} to canonical sheaves as in Serre duality results used by Pierre Deligne.
Functoriality and naturality
Ext is contravariant in its first argument and covariant in its second, with functoriality encoded by natural transformations studied by Eilenberg and Mac Lane; Yoneda composition yields products Ext^i(A,B) × Ext^j(B,C) → Ext^{i+j}(A,C), a composition law central to the work of Nicolas Bourbaki and Saunders Mac Lane. Natural isomorphisms such as Hom-tensor adjunctions relate Ext to Tor via universal coefficient phenomena explored by Eilenberg, Mac Lane, and Jean Leray. These naturality properties underpin coherence theorems in the Derived category developed by Verdier and expanded by Grothendieck.
Applications and relations to other homological invariants
Ext interrelates with numerous invariants: Tor groups in algebraic topology as in Edwin Spanier’s texts; group cohomology by Jean-Pierre Serre and John Tate; local cohomology developed by Alexander Grothendieck; and deformation theory as studied by Michael Artin, Alexander Grothendieck, and Maurice Auslander. In representation theory, Ext detects extensions of representations in the work of Issai Schur and Claude Chevalley; homological dimensions like projective dimension and injective dimension are measured via vanishing of Ext groups in the manner of Auslander–Buchsbaum and Bass results. Relations with K-theory and index theorems involve contributions from Alain Connes, Jean-Pierre Serre, and Alexander Grothendieck.