Hochschild

Hochschild is a surname of German origin. In scholarly contexts, the name is most prominently associated with the mathematician Gerhard Hochschild, whose work in algebra led to the development of Hochschild cohomology and Hochschild homology, foundational concepts in modern algebra.

In mathematics, Hochschild cohomology is a theory for associative algebras over a field and their bimodules. For an algebra A and an A-bimodule M, the n-th Hochschild cohomology group HH^n(A,M) is defined via the Hochschild cochain complex consisting of maps from A^{⊗ n} to M with a specific differential. The theory captures derivations, extensions, and deformations of algebras. Key low-degree groups have concrete interpretations: HH^0(A,M) consists of central invariants, HH^1 relates to derivations modulo inner derivations, and HH^2 governs infinitesimal deformations of the algebra structure. Hochschild cohomology interacts with topics in deformation theory, noncommutative geometry, and representation theory, and it generalizes classical notions of algebraic invariants.

Hochschild homology is a related construction that uses a chain complex C_n(A,M) = M ⊗ A^{⊗ n} with boundary maps to produce a sequence of groups HH_n(A,M). This dual theory provides invariants of the algebra that reflect its structure from a homological perspective and connects to cyclic homology in noncommutative geometry.

The Hochschild–Serre spectral sequence, named after Hochschild and Serre, appears in the study of group cohomology and Lie algebra cohomology. It relates the cohomology of a group extension or a Lie algebra extension to the cohomology of its constituent parts.

Beyond mathematics, Hochschild as a surname may appear in historical or genealogical contexts.