# Motivation for Koszul complex

Koszul complex is important for homological theory of commutative rings. However, it's hard to guess where it came from. What was the motivation for Koszul complex?

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In addition to the answer below. If you know about the Koszul dual of a Koszul algebra $A$, then it is isomorphic to the Ext-algebra $Ext_A(k,k)$ where $k$ is the (direct sum of distinct isoclasses of) simple module(s) of $A$. The natural computation of Ext-groups come from looking at complex, the Koszul complex is exactly (quasi-isomprhic to) the projective resolution of the simple module(s). Beilinson-Ginzburg-Soergel's paper is the place to look at this approach. –  Aaron Jun 28 '12 at 17:06
There is a related question here mathoverflow.net/questions/146353/history-of-koszul-complex, along with some interesting answers. –  user143746 Apr 17 '14 at 6:20

I don't know the historical origins, but it is not so hard to make up a story:

Consider the basic example $$0 \to k[x] \to k[x] \to k \to 0,$$ where the middle arrow is mult. by $x$. This is a resolution of $k = k[x]/(x)$ as a $k[x]$-module.

Now suppose you want to generalize this to obtain a resolution of $k = k[x_1,...,x_n]/(x_1,...,x_n)$ as a $k[x_1,...,x_n]$-module. It is not hard to see that you need "one copy" of the above sequence for each variable; tensoring these all together over $k$ gives you the usual Koszul resolution of $k$ over $k[x_1,...,x_n]$.

It is not hard to pass now to the more general context of elements $a_1,\ldots,a_n$ in a ring $A$, and to imagine the the Koszul complex of $a_1,\ldots,a_n$ will related to the module $A/(a_1,\ldots,a_n)$.

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Thanks. That makes sense. I was thinking about the defintion of Koszul complex by Cartan-Eilenberg(p.151). Let $K$ be a field. Let $A = K[x_1, ..., x_n]$ be the ring of polynomials. Let $E(y_1, ..., y_n)$ be the exterior algebra on $n$ letters over $K$. Let $M$ be a $A$-module. Cartan-Eilenberg defined Koszul complex as $M \otimes E(y_1, ..., y_n)$. –  Makoto Kato Jun 17 '12 at 6:26

In this answer I would rather focus on why is the Koszul complex so widely used. In abstract terms, the Koszul complex arises as the easiest way to combine an algebra with a coalgebra in presence of quadratic data. You can find the modern generalization of the Koszul duality described in Aaron's comment by reading

To my knowledge the Koszul complex is extremely useful because you can use it even with certain $A_\infty$-structures arising from deformation quantization of Poisson structures and you relate it to the other "most used resolution in homological algebra", i.e. the bar resolution.