Koszul complex

Notice that and are both graded algebras, therefore we can define , , . Consider the complex:

(18)

It consists of injective modules, but is it exact?

The question about exactness is a nontrivial one. It depends on .

Mathematicians have tools to address this question. (The theory of quadratic algebras.)

If (18) is exact, then the algebra is called a “Koszul algebra”. If is Koszul, then is also Koszul, and vise versa. Duality of quadratic algebras is symmetric.