nLab Koszul algebra

Definition

A positively graded algebra $A$ is called Koszul (after Jean-Louis Koszul) if every graded simple object has a projective resolution where the $i$th term is generated in grade $i$. Equivalently, this holds if the internal and homological gradings on the Ext $A^!=\Ext_A(A_0,A_0)$ coincide.

The algebra $(A^!)^{\op}$ is itself Koszul and is called the Koszul dual of $A$.