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Hochschild homology

In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings.

Version
v1 · 2026-09-28 · History
Domain-specific #
9884
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Homological Algebra → Mathematics

Core Idea

Hochschild homology is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by . where 0 is the basepoint, and the morphisms are the basepoint preserving set maps.

Scope of Application

  • Topological Hochschild homology. showed that the Hasse–Weil zeta function of a smooth proper variety over \mathbb{F}p can be expressed using regularized determinants involving topological Hochschild homology.

  • Definition of Hochschild homology of algebras. Let k be a field, A an associative k-algebra, and M an A-bimodule.

  • Definition of Hochschild homology of algebras. The enveloping algebra of A is the tensor product A^e=A\otimes A^o of A with its opposite algebra.

  • Hochschild complex. Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule.

  • Hochschild complex. We will write A^{\otimes n} for the n-fold tensor product of A over k.

Clarity

A clear use of Hochschild homology names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. The strongest recognition evidence in the frozen account is: Its product structure is given by the wedge product of vectors, so \begin{align}.

Manages Complexity

Hochschild homology compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—thus, if F is a functor F\colon \operatorname{Fin} \to k\text{-mod} , we get a simplicial module by composing F with S^1 .—and the practical consequence—the (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for \mathcal{C} = D(\mathbb{Z}).

Abstract Reasoning

  1. Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings.
  3. Check operation and conditions. A skeleton for the category of finite pointed sets is given by the objects.
  4. Demand recognition evidence. Its product structure is given by the wedge product of vectors, so \begin{align}.
  5. Test variation.

Knowledge Transfer

Within the home domain. Knowledge about Hochschild homology transfers literally when a new case preserves the same carrier type, relation, and recognition test. showed that the Hasse–Weil zeta function of a smooth proper variety over \mathbb{F}p can be expressed using regularized determinants involving topological Hochschild homology. Let k be a field, A an associative k-algebra, and M an A-bimodule. Beyond the home domain. No canonical parent is asserted for Hochschild homology.

Relationships to Other Abstractions

Local relationship map for Hochschild homologyParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Hochschild homologyDOMAINPrime abstraction: Theory — is a kind ofTheoryPRIME

Current abstraction Hochschild homology Domain-specific

Parents (1) — more general patterns this builds on

  • Hochschild homology is a kind of Theory Prime

    Hochschild homology is a strict kind of Theory: its frozen identity entails the parent's defining structure while adding domain-specific restrictions.

Hierarchy paths (2) — routes to 2 parentless roots

Neighborhood in Abstraction Space

Hochschild homology sits in a crowded region of the domain-specific corpus (30th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Structures, Groups & Operators (42 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08