# History of Koszul complex

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

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

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

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

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For your last question: note that the Lie algebras you get are pretty special since they are metabelian in the sense that the derived subalgebra is abelian. In case $R=K$ is a field and $f\neq 0$, the computation of this Lie algebra is rather straightforward: it is the Lie algebra of the group of affine homotheties of the hyperplane Ker($f$). –  YCor Oct 30 '13 at 13:19
The Koszul complex is the special case of the Chevalley-Eilenberg complex for a less weird example: the abelian Lie algebra! –  Mariano Suárez-Alvarez Oct 31 '13 at 6:16
Yves, thanks for remark on affine homotheties, it did not occur to me. But I think definition for metabelian is different: metabelian is 2-step nilpotent, not 2-step solvable. –  Sasha Pavlov Oct 31 '13 at 12:41
Marino, quantifier is missing. Koszul complex for any linear form is a Chevalley-Eilenberg complex for Lie algebra of that form. –  Sasha Pavlov Oct 31 '13 at 12:45
@Sasha Pavlov: there are two incompatible definitions of metabelian in the air. In group theory it essentially always mean that the derived subgroup is abelian. Possibly Lie algebra theory, I think both definitions are still in use (see arxiv.org/pdf/1302.0825v1.pdf, the first given by google for "metabelian lie algebra", for an example where it means 2-step solvable). –  YCor Nov 1 '13 at 14:20

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.