Koszul algebra
In abstract algebra, a Koszul algebra R {\displaystyle R} is a graded k {\displaystyle k} -algebra over which the ground field k {\displaystyle k} has a linear minimal graded free resolution, i.e., there exists an exact sequence: ⋯ → ( R ( − i ) ) b i → ⋯ → ( R ( − 2 ) ) b 2 → ( R ( − 1 ) ) b 1 → R → k → 0. {\displaystyle \cdots \rightarrow (R(-i))^{b_{i}}\rightarrow \cdots \rightarrow (R(-2))^{b_{2}}\rightarrow (R(-1))^{b_{1}}\rightarrow R\rightarrow k\rightarrow 0.} for some nonnegative integers b i {\displaystyle b_{i}} .