nLab
Chevalley-Eilenberg chain complex

Given a $k$-Lie algebra $𝔤$ over a commutative unital ring $k$ which is free as a $k$-module, the Chevalley–Eilenberg chain complex is a particular projective resolution $V_{*}(𝔤)\to k$ of the trivial $𝔤$-module $k$ in the abelian category of $𝔤$-modules (what is the same as $U𝔤$-modules, where $U𝔤$ is the universal enveloping algera of $𝔤$). Graded components of the underlying $k$-module this resolution is given by

$$V_p(𝔤)=U(𝔤){\otimes }_k{\Lambda }^p𝔤$$V_p(\mathfrak{g}) = U(\mathfrak{g})\otimes_k \Lambda^p{\mathfrak{g}}

and it has the obvious $U𝔤$-module structure by multiplication in the first tensor factor, because ${\Lambda }^p𝔤$ is free as a $k$-module.

If $u\in U𝔤$ and $x_i\in 𝔤$ then the differnetial is given by

$$d(u\otimes x_1\wedge \cdots \wedge x_p)=$$d(u\otimes x_1 \wedge \cdots \wedge x_p) =

$$=\sum _{i=1}^p(-1{)}^{i+1}{\mathrm{ux}}_i\otimes x_1\wedge \cdots \wedge {\hat{x}}_i\wedge \cdots \wedge x_p+\sum _{i<j}(-1{)}^{i+j}u\otimes [x_i,x_j]\wedge \cdots \wedge {\hat{x}}_i\cdots \wedge {\hat{x}}_j\cdots \wedge x_p$$= \sum_{i = 1}^p (-1)^{i+1} ux_i \otimes x_1 \wedge \cdots \wedge \hat{x}_i\wedge \cdots \wedge x_p + \sum_{i\lt j} (-1)^{i+j} u\otimes [x_i, x_j] \wedge \cdots \wedge \hat{x}_i\cdots \wedge \hat{x}_j\cdots \wedge x_p

See also Lie algebra homology, Lie algebra cohomology, Chevalley–Eilenberg cochain complex and Chevalley–Eilenberg algebra.