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Hochschild HomologyEdit

Hochschild homology is a homology theory for associative algebras that attaches a sequence of abelian groups or modules to an algebra A, encoding information about how A sits inside its bimodule structure and how A can be deformed or compared to simpler models. Introduced by Bernhard Hochschild in the mid-20th century, the theory has grown into a central tool in areas ranging from Algebra and Representation theory to Algebraic geometry and Noncommutative geometry. The construction uses the Hochschild chain complex built from tensor powers of A, and its homology groups HH_n(A) carry rich arithmetic and geometric content. In the commutative, smooth case over a field of characteristic zero, HH_*(A) recovers the familiar differential forms via the Hochschild–Kostant–Rosenberg theorem, making Hochschild homology a natural noncommutative analogue of de Rham cohomology.

In modern practice, Hochschild homology sits alongside related invariants such as Hochschild cohomology and cyclic homology, forming part of a broader framework that connects homological algebra with geometry, deformation theory, and mathematical physics. The invariant is functorial in A and interacts with base changes, stacky geometry, and the derived categories that appear in DG algebra and A-infinity algebra.

Overview

Hochschild homology associates to an associative algebra A over a base ring or field k a chain complex C_*(A) whose n-th term is A ⊗ A ⊗ ... ⊗ A (n+1 factors) with a boundary map b defined by the product in A and cyclic permutations. The homology of this complex is HH_n(A). This construction generalizes familiar ideas from the study of differential forms and traces and specializes in important ways when A is commutative or finite-dimensional.

A guiding intuition is that Hochschild homology behaves like a theory of differential forms for noncommutative spaces, with HH_0(A) measuring a kind of “trace space” and higher HH_n(A) capturing higher-order relations among elements of A. In the commutative, smooth setting, the HKR theorem identifies HH_n(A) with Ω^n_{A/k}, the module of differential n-forms on Spec(A). This identification links algebraic and geometric viewpoints and provides a bridge to classical de Rham theory.

The relationship between Hochschild homology and cyclic homology broadens the perspective further. Cyclic homology HC_(A) and periodic cyclic homology HP_(A) refine HH_*(A) to account for traces and cyclic symmetries, echoing the role of traces in index theory. In noncommutative geometry, these invariants play a role analogous to characteristic classes and integration on manifolds, connecting to Connes’ noncommutative differential geometry and to K-theory via appropriate pairings.

For those looking at computational and structural aspects, Hochschild homology is compatible with many constructions, including base change and localization, and it interacts with derived categories, making it a central tool in modern homological algebra.

Construction and basic properties

  • The Hochschild chain complex C_n(A) is built from tensor powers of A: C_n(A) = A ⊗ A ⊗ ... ⊗ A (n+1 factors).
  • The boundary map b uses the multiplication in A and a cyclic permutation of the tensor factors. Explicitly, for a_0 ⊗ a_1 ⊗ ... ⊗ a_n, b(a_0 ⊗ ... ⊗ a_n) = a_0a_1 ⊗ a_2 ⊗ ... ⊗ a_n + sum of other terms + (−1)^n a_n a_0 ⊗ a_1 ⊗ ... ⊗ a_{n-1}.
  • The n-th Hochschild homology group is HH_n(A) = H_n(C_*(A), b).
  • The construction is functorial in A: a ring homomorphism A → B induces a map HH_(A) → HH_(B).

For a base ring k, one often considers A as a k-algebra and discusses HH_*(A/k) to emphasize the role of the base, with the HKR comparison providing a precise bridge to differential forms when A is smooth over k.

Examples

  • If A = k is just the base field, then HH_0(A) ≅ k and HH_n(A) = 0 for n > 0.
  • If A = k[x], the polynomial ring in one variable over k and k has characteristic zero, then HH_n(A) ≅ Ω^n_{A/k} via the HKR theorem. In concrete terms, HH_0(k[x]) ≅ k[x], HH_1(k[x]) ≅ k[x] dx, and higher forms correspond to higher differential forms on the affine line.
  • For noncommutative examples, such as A = k⟨x,y⟩/(relations), HH_*(A) can be computed using standard resolutions (e.g., the bar resolution) and reflects how the algebra fails to be commutative, with HH_n(A) capturing obstructions to commutativity in a homological sense.

Variants and connections

  • Hochschild cohomology HH^(A) classifies infinitesimal deformations of A, with the Gerstenhaber bracket giving the algebraic structure of a graded Lie algebra. This makes HH^(A) central to deformation theory and to understanding how algebras deform within families.
  • The Hochschild–Kostant–Rosenberg theorem provides a sharp description of HH_*(A) for smooth commutative algebras, tying Hochschild homology to the algebra of differential forms. This result highlights the conceptual link between noncommutative invariants and classical geometry.
  • Cyclic homology HC_(A) extends Hochschild homology by incorporating a cyclic symmetry, leading to a long exact sequence that parallels Connes’ periodicity map. Periodic cyclic homology HP_(A) is particularly important in index theory and in comparisons with topological invariants.
  • Topological Hochschild homology THH(A) generalizes the construction to ring spectra in stable homotopy theory, connecting algebraic invariants to homotopical methods and to topological methods in algebraic K-theory.
  • The broader ecosystem includes relationships with Derived categorys, DG algebra, and A-infinity algebra, where Hochschild invariants are computed in derived or higher-categorical contexts.
  • In noncommutative geometry, Hochschild homology plays a role as a recipient of Chern character maps from K-theory and as a bridge to geometric notions on noncommutative spaces.

Computation and methods

  • One computes HH_(A) by resolving A as a bimodule over itself and applying the bar resolution or other projective resolutions to obtain a chain complex whose homology yields HH_(A).
  • Spectral sequences, such as the Hochschild–Serre spectral sequence, aid in computations when A has a compatible filtration or a fibration-like decomposition.
  • For many naturally occurring algebras (e.g., finite-dimensional algebras, function algebras on smooth varieties), the HKR theorem gives a direct link to differential forms, simplifying calculations in the smooth commutative setting.
  • Software and computational tools in homological algebra can assist with explicit calculations for specific algebras, especially when working over a fixed base field and dealing with finite-dimensional cases.

Generalizations and related theories

  • Extensions to non-smooth or noncommutative contexts require more sophisticated machinery, and the invariants can detect singularities or intricate bimodule structures.
  • Connections to deformation theory, index theory, and noncommutative geometry motivate studying Hochschild homology in tandem with HH^(A), HC_(A), and HP_*(A), as well as with their topological analogues.
  • The interplay with differential forms in the commutative case provides a template for thinking about geometric content in noncommutative settings, and the broad framework invites applications to representation theory, algebraic geometry, and mathematical physics.

See also