History of Koszul complex

Question

This is a question about the history of commutative algebra. I'm curious why the Koszul complex from commutative algebra is called the Koszul complex? All of Koszul's early papers are about Lie algebras and Lie groups, in particular about the Chevalley-Eilenberg complex. He never published papers on commutative algebra. It looks like the Koszul complex (under this name) was first used in early versions of Serre's work on multiplicities around 1950. What was the motivation to associate this complex with Koszul?

It is clear to me that that any linear form $f$ on a module $M$ over a commutative ring $R$ gives a Lie algebra structure on $M$ with bracket $[x,y]=f(x)y-f(y)x$. The Chevalley-Eilenberg complex (with coefficients in the trivial representation) of this Lie algebra is the Koszul complex of $f$. So the Koszul complex is a special case of the Chevalley-Eilenberg complex for one stupid Lie algebra.

Was this observation a bridge between Lie algebra cohomology and commutative algebra that actually gave us the Koszul complex around 1950?

By the way, is there a special name for such Lie algebras associated to a form?

Answer 1

Although the germ of the idea might've appeared in Koszul's earlier work on the cohomology of Lie algebras and homogeneous spaces, it seems that the first full-fledged appearance of the Koszul complex/resolution is in Koszul, Sur un type d'algèbres différentielles en rapport avec la transgression, Colloque de topologie (espaces fibrés), Bruxelles (1950), 73–81. The primary motivation there is topological/geometric (cohomology of fiber bundles), but Koszul does give fairly abstract algebraic results and definitions. Specifically, consider a principal $G$-bundle $p\colon E\to B$, where $G$ is a compact connected Lie group. Write $x_1,\ldots,x_l$ for the primitive generators of $H^\ast(G)$, so that $H^\ast(G)=\bigwedge^l_{i=1} x_i$. Then there are $G$-invariant differential forms $\{\omega_i\}$ on $E$ whose restrictions $\{\xi_i\}$ to a fiber $G$ are bi-invariant forms that represent the classes $\{x_i\}$ and such that $d\xi_i$ is the image under $p^\ast$ of some form $c_i$ on the base $B$. The exterior algebra $\Omega^\ast(B)$ of $B$ may be viewed as a module over the polynomial ring $A=\mathbb R[c_1,\ldots,c_l]$. Koszul is led to the "Koszul complex" $$ {\textstyle \bigwedge_{i=1}^l} x_i \otimes \Omega^\ast(B) $$ (with the appropriate differential) through topological considerations: he notes that in certain cases (for nice enough $B$), one can replace $\Omega^\ast(B)$ above with $H^\ast(B)$ and then the resulting complex $\bigwedge x_i \otimes H^\ast(B)= H^\ast(G) \otimes H^\ast(B)$ computes the cohomology of $E$.

Koszul takes a look at the general properties of "Koszul complexes" of the form $E \otimes M$ where $E=\bigwedge_{i=1}^l x_i$ and $M$ is a module over $A=k[x_1,\ldots,x_l]$, and calls the resulting cohomology $H^\ast(M)$ the cohomology of the $A$-module $M$. He proceeds to use this machinery to give a generalization of Hilbert's syzygy theorem. This is, e.g., the context in which the Koszul complex arises in Cartan & Eilenberg's book on homological algebra (see Ch. VIII, sections 4 and 6)---and this is probably (?) the first textbook appearance of the construction.

See also A. Haefliger, Des espaces homogènes à la résolution de Koszul, Ann. Inst. Fourier 37(4) (1987), 5–13, for some interesting historical commentary on Koszul's work.