 Koszul complex with module coefficients

Tensor the finite free Koszul complex on $f_1,\dots ,f_r$ with the module $M$. For a regular sequence on a module, its positive Koszul homology vanishes. If the sequence is regular on $R$, this complex instead computes ${\mathrm{Tor}}^R(R/(f),M)$ for arbitrary $M$.