For the $R$-$R$-<bimodule> $M$, the <Hochschild chain complex> is
$$
C_n(R,M)=M\otimes_kR^{\otimes_kn},
$$
with boundary
$$
\begin{aligned}
b(m\otimes r_1\otimes\cdots\otimes r_n)
={}&mr_1\otimes r_2\otimes\cdots\otimes r_n\\
&+\sum_{i=1}^{n-1}(-1)^im\otimes r_1\otimes\cdots\otimes r_ir_{i+1}\otimes\cdots\otimes r_n\\
&+(-1)^nr_nm\otimes r_1\otimes\cdots\otimes r_{n-1}.
\end{aligned}
$$
Then
$$
HH_n(R,M)=H_n(C_\bullet(R,M),b).
$$
The <Hochschild cochain complex> is $C^n(R,M)=\operatorname{Hom}_k(R^{\otimes_kn},M)$ with
$$
\begin{aligned}
(\delta f)(r_1,\ldots,r_{n+1})
={}&r_1f(r_2,\ldots,r_{n+1})\\
&+\sum_{i=1}^{n}(-1)^if(r_1,\ldots,r_ir_{i+1},\ldots,r_{n+1})\\
&+(-1)^{n+1}f(r_1,\ldots,r_n)r_{n+1},
\end{aligned}
$$
and $HH^n(R,M)=H^n(C^\bullet(R,M),\delta)$.

A <derivation into a bimodule> is a $k$-linear map $d:R\to M$ satisfying
$$
d(rs)=r d(s)+d(r)s.
$$
It is an <inner derivation> when $d=d_m$ for some $m\in M$, where $d_m(r)=rm-mr$. The degree-one cocycle equation is precisely the <Leibniz rule>, and the degree-one coboundaries are the inner derivations. Therefore the <First Hochschild cohomology as outer derivations> is
$$
\boxed{HH^1(R,M)\cong\operatorname{Der}_k(R,M)/\operatorname{Inn}(R,M).}
$$

Let $\mu:R\otimes_kR\to R$ be multiplication and $\Omega=\ker\mu$, equipped with the outer bimodule structure. The <universal bimodule derivation>
$$
D:R\longrightarrow\Omega,\qquad D(r)=r\otimes1-1\otimes r
$$
satisfies $D(rs)=rD(s)+D(r)s$. Composition with $D$ defines
$$
\operatorname{Hom}_{R-R}(\Omega,M)\longrightarrow\operatorname{Der}_k(R,M).
$$
For a derivation $d$, its inverse image under this map is
$$
\theta_d\left(\sum_ir_i\otimes s_i\right)=\sum_id(r_i)s_i.
$$
If $\sum_ir_is_i=0$, the Leibniz rule shows that this formula is both left and right $R$-linear. Moreover every element of $\Omega$ is $\sum_iD(r_i)s_i$, proving existence and uniqueness.

For $d_m(r)=rm-mr$, the corresponding map is
$$
\boxed{\theta_m\left(\sum_ir_i\otimes s_i\right)=\sum_ir_ims_i.}
$$
These are exactly the maps $\Omega\to M$ which extend to bimodule maps $R\otimes_kR\to M$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 148 / 5 / Solution

{c}
{title2=$HH^1(R,M)=\operatorname{Der}_k(R,M)/\operatorname{Inn}(R,M)$}

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
$$
HH^1(R,M)\cong\operatorname{Der}_k(R,M)/\operatorname{Inn}(R,M).
$$

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 Incoming links: First Hochschild cohomology as outer derivations