Koszul homology
= Koszul homology
{c}
{title2=$H_i(f;M)$}
Koszul homology is the homology of the <Koszul complex with module coefficients>. Its degree-zero term is $M/(f)M$, and its positive terms record failures of regularity. Each generator acts trivially on homology by the <Koszul homotopy for multiplication by a generator>.