Hochschild homology

Hochschild homology
Name	Hochschild homology
Field	Algebra, Algebraic topology, Noncommutative geometry
Introduced	1945
Introduced by	Gerhard Hochschild

Hochschild homology is an invariant of associative algebras that captures multilinear and cyclic information about algebraic structures and their modules, connecting algebraic, geometric, and topological frameworks. It arose in mid‑20th century work on homological algebra and representation theory and has since interfaced with deformation theory, noncommutative geometry, and index theory. The theory ties into algebraic K‑theory, cyclic homology, and various cohomological techniques developed in the 20th and 21st centuries.

Definition and basic properties

Hochschild homology is defined for an associative algebra A over a commutative ring k and a bimodule M, producing homology groups HH_n(A,M) that generalize derived functors and Tor groups; key early contributors include Gerhard Hochschild, Samuel Eilenberg, Saunders Mac Lane, Jean-Pierre Serre, and Israel Gelfand. Fundamental properties mirror those of Tor and Ext: functoriality under algebra maps and bimodule maps, long exact sequences associated to short exact sequences of bimodules, and base change relations studied by Alexander Grothendieck, Jean-Louis Verdier, and Alexander Beilinson. Hochschild homology satisfies dimension statements analogous to Hochschild–Kostant–Rosenberg results by Bertram Kostant and Alexandre Rosenberg, and connects to smoothness criteria considered by Pierre Deligne, Michel Artin, and Michael Atiyah. For finite dimensional algebras over fields studied by Issai Schur and Richard Brauer, Hochschild homology reflects Morita invariance results related to work of Hyman Bass and Henri Cartan.

Hochschild complex and chain models

The Hochschild complex C_*(A,M) is constructed from tensor powers A^{\otimes n} and the module M with a boundary operator encoding the algebra multiplication; this combinatorial chain model was formalized in the work of Eilenberg and Mac Lane and later reinterpreted using simplicial and cyclic objects by Jean-Louis Loday, Alain Connes, and André Joyal. Alternative models include bar complexes and normalized bar resolutions used by Samuel Eilenberg and John C. Moore, and differential graded algebra approaches developed in the context of Maxim Kontsevich's work on deformation quantization and Dennis Sullivan's rational homotopy theory. Operadic and model category formulations due to Markl, Berger, and Vladimir Hinich provide homotopical control over multiplicative and higher structure, while mixed complexes and bicomplex perspectives used by Connes and Loday lead to spectral sequences connecting Hochschild homology to cyclic and periodic homology studied by Jean-Michel Bismut.

Computations and examples

Classical computations include Hochschild homology of polynomial algebras, group algebras, and matrix algebras: polynomial algebra computations relate to differential forms explored by Kostant and Rosenberg, group algebra computations connect to group cohomology studied by Henri Cartan and Jean-Pierre Serre, and Morita invariance for matrix algebras follows from results of Bass and Cartan. Calculations for local algebras and complete intersections tie into results by David Eisenbud and Serre on homological dimensions, while computations for path algebras of quivers exploit work by Pierre Gabriel and Idun Reiten. Examples from algebraic geometry include Hochschild homology of smooth projective varieties where comparisons to Hodge theory invoke Phillip Griffiths, Joseph Harris, and the Hochschild–Kostant–Rosenberg theorem used in studies by Maxim Kontsevich and Alexander Grothendieck. Explicit low‑dimensional computations for surface algebras, quantum groups influenced by Vladimir Drinfeld and Michio Jimbo, and orbifold algebras considered by William Thurston illustrate the breadth of examples.

Relationship with cyclic homology and Connes' long exact sequence

Hochschild homology embeds into cyclic homology via Connes' SBI (or periodicity) sequence introduced by Alain Connes and systematized by Jean-Louis Loday, generating a long exact sequence that relates Hochschild homology HH_*(A) to cyclic HC_*(A) and periodic cyclic HP_*(A); this relation underpins comparisons with de Rham and Hodge theories developed by Grothendieck and Deligne. The Connes operator B and the periodicity operator S, central to the SBI sequence, play roles in index theorems linked to the Atiyah–Singer index theorem established by Michael Atiyah and Isadore Singer, and in noncommutative geometry frameworks applied by Connes to foliations and operator algebras studied by John von Neumann and Alain Bost. The sequence provides computational tools exploited in algebraic K‑theory contexts advanced by Daniel Quillen, Charles Weibel, and Hyman Bass.

Applications in algebra, geometry, and topology

Hochschild homology appears in deformation theory and obstruction calculus in the work of Maurice Gerstenhaber and Jim Stasheff, in the classification of formal deformations inspired by Kontsevich's deformation quantization, and in derived categories and Fourier–Mukai transforms used by Alexander Bondal and Dmitri Orlov. In algebraic geometry, it links to Hodge theory and the study of singularities pursued by John Milnor and Bernard Teissier; in topology it connects to free loop space homology and string topology developed by Chas and Sullivan, and to factorization homology concerns explored by Kevin Costello and Owen Gwilliam. Applications to mathematical physics include connections with conformal field theory and topological quantum field theory researched by Edward Witten and Graeme Segal.

Variants and generalizations (relative, continuous, topological)

Multiple variants extend Hochschild homology: relative Hochschild homology for algebra extensions studied by Hochschild and Cartan, continuous and completed versions for topological algebras used in functional analysis contexts related to John von Neumann and C*-algebra theory pursued by Gelfand and Maurice Riesz, and topological Hochschild homology (THH) defined for ring spectra in stable homotopy theory developed by Daniel Quillen, J. Peter May, Vladimir Voevodsky, and Michael Hopkins. Cyclotomic and topological cyclic homology (TC) created by Bökstedt, Hsiang, and Madsen refines THH for applications to algebraic K‑theory and arithmetic geometry studied by Kazuya Kato and Luc Illusie, while relative and equivariant formulations draw on group actions considered by Galois and Emmy Noether in classical algebraic contexts.

Category:Homological algebra