Koszul homotopy for multiplication by a generator ($\partial h_j+h_j\partial =f_j\mathrm{id}$)

Exterior multiplication $h_j(u)=e_j\wedge u$ is a chain homotopy between multiplication by $f_j$ and zero on a Koszul complex. A linear combination of these homotopies is a contracting homotopy when the generators span the unit ideal.