Hochschild cochain complex

The Hochschild cochains are $C^n(R,M)={\mathrm{Hom}}_k(R^{{\otimes }_{k}n},M)$, with coboundary

The Hochschild cohomology groups are

A $k$-linear derivation from a $k$-algebra $R$ to an $R$-$R$-bimodule $M$ is a map $d:R\to M$ satisfying

For $m\in M$, the formula $d_m(r)=rm-mr$ defines an inner derivation $R\to M$.

The degree-one Hochschild cocycles are exactly the maps that are a derivation into a bimodule, while degree-one coboundaries are exactly the inner derivations. Therefore

Let $\mu :R{\otimes }_{k}R\to R$ be multiplication and ${\Omega }_R^{\mathrm{nc}}=\ker \mu$, with the outer $R$-bimodule structure. Thenis a derivation and composition with $D$ gives a natural isomorphismThe inverse sends $d$ toThe condition ${\sum }_i{r}_i{s}_i=0$ makes this map left as well as right $R$-linear. Every element of ${\Omega }_R^{\mathrm{nc}}$ is a sum of terms $D(r)s$, which proves uniqueness.

Under the universal isomorphism, the inner derivation $d_m(r)=rm-mr$ corresponds toThus inner derivations correspond exactly to bimodule maps on ${\Omega }_R^{\mathrm{nc}}$ that extend across the inclusion ${\Omega }_R^{\mathrm{nc}}\subseteq R{\otimes }_{k}R$.