Self-duality of the Koszul complex (${\mathrm{Hom}}_R(K_i,R)\cong K_{r-i}$)

The perfect exterior pairing in complementary degrees identifies the dual of a Koszul complex with its degree reversal, with suitable differential signs. For a regular sequence, its Koszul resolution consequently gives ${\mathrm{Ext}}_R^i(R/(f),R)=0$ except in degree $r$, where it is $R/(f)$.