---
tag: 676K167XS8
kind: concept
name: Cyclic homology
msc: ["16E40", 19D55]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_homology
same_as:
  wikidata: Q939179
  wikipedia: Cyclic homology
---

## Summary

Cyclic homology is a homology theory for [[associative algebras|3CTW1P4HBP]] introduced by Alain Connes and, independently, by Boris Tsygan.
It refines [[Hochschild homology|S5H153HFWC]] by taking into account the cyclic rotation of tensor factors.
For smooth commutative algebras in characteristic $0$ it is computed from de Rham cohomology.
So it plays the role of de Rham theory in noncommutative geometry.
It is also the target of trace maps from [[algebraic K-theory|5GVFZWMWC8]].

## Explanations

### L1 human

Hochschild homology is built from strings $a_0 \otimes a_1 \otimes \cdots \otimes a_n$ of elements of an algebra.
Cyclic homology treats such a string as written around a circle, so that rotating it gives the same object up to a sign.
This extra symmetry makes cyclic homology closer to topology and to de Rham cohomology.
For the ground field $k$ itself, cyclic homology is $k$ in every even degree and $0$ in odd degrees.
Connes used it to define index-type invariants for noncommutative spaces.