  We introduce and study the Koszul complex for a Hecke $R$-matrix. Its
cohomologies, called the Berezinian, are used to define quantum
superdeterminant for a Hecke $R$-matrix. Their behaviour with respect to Hecke
sum of $R$-matrices is studied. Given a Hecke $R$-matrix in $n$-dimensional
vector space, we construct a Hecke $R$-matrix in $2n$-dimensional vector space
commuting with a differential. The notion of a quantum differential supergroup
is derived. Its algebra of functions is a differential coquasitriangular Hopf
algebra, having the usual algebra of differential forms as a quotient. Examples
of superdeterminants related to these algebras are calculated. Several remarks
about Woronowicz's theory are made.
