Koszul complex on central ring elements
= Koszul complex on central ring elements
{c}
{title2=$K_k=\bigoplus_{|I|=k}R e_I$}
For a possibly noncommutative ring and central $t_i$, the formal exterior basis gives free <bimodules> with differential $d(e_{i_1}\wedge\cdots\wedge e_{i_k})=\sum_a(-1)^{a-1}t_{i_a}e_{i_1}\wedge\cdots\widehat{e_{i_a}}\cdots\wedge e_{i_k}$. Centrality makes this a bimodule differential. Pairwise cancellation proves $d^2=0$.