Bar resolution

Summary

The bar resolution of a group $G$ is an explicit free resolution of the trivial module $\Z$ over the group ring $\Z[G]$. Its basis elements in degree $n$ are the symbols $[g_1 | \cdots | g_n]$, one for each list of $n$ elements of $G$. It turns group cohomology into a concrete computation with functions of $n$ group elements, which gives the classical cocycle descriptions of $H^1$ and $H^2$. The same construction exists for augmented algebras, where it computes Hochschild homology and Tor.