Koszul complex with module coefficients
= Koszul complex with module coefficients
{c}
{title2=$K(f;M)=K(f;R)\otimes_RM$}
Tensor the finite free <Koszul complex> on $f_1,\ldots,f_r$ 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 $\operatorname{Tor}^R(R/(f),M)$ for arbitrary $M$.