Koszul algebra

Remarks

Example: $A = k[x]$ with $\deg x = 1$. The resolution $0 \to A(-1) \xrightarrow{x} A \to k \to 0$ has $P_0 = A$ generated in degree $0$ and $P_1 = A(-1)$ generated in degree $1$. More generally, the Koszul complex is a linear resolution of $k$ over $k[x_1, \dots, x_n]$. Non-example: $A = k[x]/(x^3)$ with $\deg x = 1$. The minimal resolution starts with $P_0 = A$ and $P_1 = A(-1)$, mapping the generator to $x$. The kernel of multiplication by $x$ on $A$ is spanned by $x^2$, so $P_2$ is generated in degree $1 + 2 = 3$, not $2$. Equivalent form: $A$ is Koszul if and only if $\operatorname{Ext}^i_A(k, k)$ is concentrated in internal degree $i$ for every $i$. Every Koszul algebra is quadratic: it is generated by $A_1$ with relations in degree $2$.