  We construct the super Koszul complex of a free supercommutative $A$-module
$V$ of rank $p|q$ and prove that its homology is concentrated in a single
degree and it yields an exact resolution of $A$. We then study the dual of the
super Koszul complex and show that its homology is concentrated in a single
degree as well and isomorphic to $\Pi^{p+q} A$, with $\Pi$ the parity changing
functor. Finally, we show that, given an automorphism of $V$, the induced
transformation on the only non-trivial homology class of the dual of the super
Koszul complex is given by the multiplication by the Berezinian of the
automorphism, thus relating this homology group with the Berezinian module of
$V$.
