##  [Hochschild–Kostant–Rosenberg Theorem](/hochschild-kostant-rosenberg-theorem) 

  ##  [Hochschild–Kostant–Rosenberg Theorem](https://puremath.quantumdictionary.io/hochschild-kostant-rosenberg-theorem-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An isomorphism identifying the Hochschild homology of a smooth commutative algebra with its algebraic differential forms: for a smooth commutative k-algebra A (under the usual hypotheses, e.g.

 

 

 

 

 





 

 



 ##  [Hochschild–Kostant–Rosenberg Theorem](https://algebra.quantumdictionary.io/hochschild-kostant-rosenberg-theorem-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A theorem identifying the Hochschild homology HH_*(A) of a smooth commutative algebra A (or smooth algebraic variety) with the algebra of differential forms Ω^*_A, via the Hochschild–Kostant–Rosenberg (HKR) map, thereby linking homological algebra to differential geometry in the smooth commutative setting.