# Systematic approach to the Chevalley-Eilenberg resolution

When computing derived functors, one would like to be able to find injective and projective resolutions of objects. Two important examples come to mind where a single resolution allows one to compute (co)homology in all, cases, namely, group and Lie algebra (co)homology. In these cases, one is interested in computing the right (left) derived functors of the invariant (coinvariant) functors. The invariant functors are respectively naturally isomorphic to $$\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z},{-})\text{ and }\operatorname{Hom}_{\mathrm{U}\mathfrak{g}}(\mathbb{K},{-}),$$ where I am regarding $\mathbb{Z}$ and $\mathbb{K}$ respectively as trivial $\mathbb{Z}[G]$ and $\mathrm{U}\mathfrak{g}$ modules. (The coinvariants are naturally isomorphic to analogous tensor products, but I only need to discuss one to ask the relevant question.) Thus, if I can find projective resolutions of $\mathbb{Z}$ and $\mathbb{K}$, then I can compute the right derived functors of $\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z},M)$ and $\operatorname{Hom}_{\mathrm{U}\mathfrak{g}}(\mathbb{K},N)$ for any $G$-module $M$ and $\mathfrak{g}$-module $N$.

In fact, it turns out that, there are canonical resolutions in both cases, the Bar resolution and Chevalley-Eilenberg resolution respectively. Furthermore, these are of course not the only examples of canonical resolutions existing 'in nature'. This suggests that such resolutions don't just come out of thin air, but that there is a method to the madness. Indeed, Weibel discusses this (Section 8.6 Canonical Resolutions) and explains how the Bar resolution arises from a comonad (the forgetful functor from $G$-modules to Abelian groups and its left adjoint). Alas, though I see some similarity between the two, I do not know how to obtain the Chevalley-Eilenberg resolution (or the related Koszul resolution) in this way.

Is it possible to understand the Chevalley-Eilenberg resolution as something more than just a 'trick' for finding a projective resolution of $\mathbb{K}$?

• If you think of $U(\mathfrak{g})$ as a deformation of the symmetric algebra $S(\mathfrak{g})$, the Chevalley-Eilenberg resolution can be thought of as a deformation of the Koszul resolution, to which it reduces in the special case that the Lie algebra is abelian. – Qiaochu Yuan Apr 19 '17 at 3:34
• You should read about Koszul algebras. They have a canonical resolution, and they admit certain "PBW deformations" which come with corresponding deformations of that canonical resolution. Polynomial algebras are Koszul, and their PBW deformations are Lie algebras, and the deformed resolution is the C-E complex. This is one level of generality at which you can view the C-E resolution as a special case. – Mariano Suárez-Álvarez Apr 19 '17 at 3:36