Hochschild cohomology
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| Hochschild cohomology | |
|---|---|
| Name | Hochschild cohomology |
| Field | Mathematics |
| Introduced | 1945 |
| Introduced by | Gerhard Hochschild |
- Definition and basic properties
- Hochschild homology and cohomology complexes
- Algebraic structures and operations
- Computations and examples
- Relations to deformation theory and Gerstenhaber algebra
- Connections to cyclic cohomology and noncommutative geometry
- Applications and advanced topics including derived categories
Hochschild cohomology Hochschild cohomology is an invariant in algebra introduced by Gerhard Hochschild that assigns graded vector spaces to associative algebras and their bimodules; it plays a central role in the study of deformations, extensions, and derived categories associated to algebras. It connects classical results of Emmy Noether-inspired algebraic structures with later developments by researchers associated to Alexander Grothendieck, Jean-Louis Loday, and Maxim Kontsevich, and it appears in interactions with subjects influenced by Andrei Kolmogorov, David Hilbert, and Élie Cartan.
Definition and basic properties
The cohomology groups are defined for an associative algebra A over a field k and an A-bimodule M via Ext-groups computed in the category of A-bimodules; foundational constructions were developed in tandem with work of Samuel Eilenberg, Saunders Mac Lane, and Nathan Jacobson. The low-degree groups have classical interpretations: H^0 corresponds to center-like invariants related to ideas from Évariste Galois-centric symmetry, H^1 classifies derivations connected to techniques used by Niels Henrik Abel and Arthur Cayley, and H^2 parameterizes equivalence classes of algebra extensions reminiscent of results by Hermann Weyl and Richard Dedekind. Functoriality properties echo categorical themes from Alexander Grothendieck and homological patterns investigated by Jean-Pierre Serre and Henri Cartan.
Hochschild homology and cohomology complexes
The Hochschild cochain complex C^*(A,M) and chain complex C_*(A,M) are built from tensor powers A^{\otimes n} and maps akin to constructions employed by Emmy Noether and Emil Artin; their differentials are analogues of boundary operators studied by James Clerk Maxwell-era formalists and later formalized by Samuel Eilenberg and Saunders Mac Lane. Quasi-isomorphism classes of these complexes are invariants under Morita equivalence introduced by Kiiti Morita, and their derived functor realizations utilize techniques from Bernard Dwork-style homological algebra and the derived category framework of Alexandre Grothendieck and Jean-Louis Verdier. Spectral sequences used to compute these complexes owe methodological roots to work of Jean Leray and Jean-Louis Koszul.
Algebraic structures and operations
Hochschild cohomology carries a graded algebra structure with the cup product analogous to products in work by Niels Henrik Abel and graded Lie structures forming Gerstenhaber algebras introduced by Murray Gerstenhaber; these structures mirror bracket operations considered by Sophus Lie and cohomological operations studied by Emmy Noether. Operations such as the Connes B-operator and the cyclic shuffle operator relate to constructions by Alain Connes and to cyclic cohomology methods developed alongside contributions from Maxim Kontsevich and Jean-Louis Loday. Deligne’s conjecture linking operadic actions to Hochschild cochains was formulated in the milieu of ideas promoted by Pierre Deligne and pursued by researchers associated to Vladimir Drinfeld and Dennis Sullivan.
Computations and examples
Explicit calculations for group algebras of finite groups trace back to methods used by Ira Gessel-adjacent combinatorialists and algebraists inspired by Emil Artin; examples include computations for matrix algebras leveraging Morita equivalence as in works related to Ferdinand Frobenius and Richard Brauer. Polynomial algebras, exterior algebras, and path algebras of quivers yield models studied by researchers in the lineage of Pierre Gabriel, Bernhard Riemann-style geometry, and Claude Chevalley. Calculations for local algebras connect to deformation computations carried out in seminars influenced by Alexander Grothendieck and computational approaches developed by John Tate and Miles Reid.
Relations to deformation theory and Gerstenhaber algebra
The interpretation of H^2 as classifying infinitesimal deformations follows the paradigm established by Kodaira–Spencer theory and echoes deformation themes advanced by Michael Artin and Murray Gerstenhaber; these ideas link to moduli problems investigated by Alexander Grothendieck, Pierre Deligne, and Maxim Kontsevich. Formality theorems that identify Hochschild cochains with polyvector fields draw upon landmark results associated to Maxim Kontsevich and techniques parallel to those in Edward Witten-inspired topological field theory. Obstruction theories and higher-order deformations utilize homotopy algebraic structures that have been informed by research from groups around Jim Stasheff, Vladimir Drinfeld, and Dennis Sullivan.
Connections to cyclic cohomology and noncommutative geometry
Cyclic cohomology, developed notably by Alain Connes, interacts with Hochschild theory through Connes’ long exact sequence linking Hochschild and cyclic groups; these relationships feed into noncommutative geometry programs connected to researchers at institutions such as Institut des Hautes Études Scientifiques and influenced by perspectives from Pierre Deligne and Maxim Kontsevich. Index theorems for noncommutative spaces and pairings with K-theory reflect concepts propagated by Alain Connes and later work by Nigel Higson and John Roe. Connections to operator algebras and C*-algebras draw on themes from George Mackey-inspired representation theory and contributions from Alain Connes and Daniel Kastler.
Applications and advanced topics including derived categories
Hochschild cohomology is central in the theory of derived categories of algebras and schemes as advanced by Alexandre Grothendieck, Amnon Neeman, and Maxim Kontsevich; it appears in formality results, mirror symmetry statements influenced by Mikhail Gromov and Maxim Kontsevich, and in bridging algebraic and symplectic geometry as in programs advocated by Paul Seidel and Richard Thomas. Its role in categorical invariants links to work of Joseph Bernstein, Vladimir Drinfeld, and Pierre Deligne, and it provides tools for studying singularities in algebraic varieties as in investigations by Miles Reid and Markushevich. Recent advances connect Hochschild methods with homotopical algebra and infinity-categories developed in frameworks by Jacob Lurie, Clark Barwick, and André Joyal.