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We introduce and study the Koszul complex for a Hecke $R$-matrix. Itscohomologies, called the Berezinian, are used to define quantumsuperdeterminant for a Hecke $R$-matrix. Their behaviour with respect to Heckesum of $R$-matrices is studied. Given a Hecke $R$-matrix in $n$-dimensionalvector space, we construct a Hecke $R$-matrix in $2n$-dimensional vector spacecommuting with a differential. The notion of a quantum differential supergroupis derived. Its algebra of functions is a differential coquasitriangular Hopfalgebra, having the usual algebra of differential forms as a quotient. Examplesof superdeterminants related to these algebras are calculated. Several remarksabout Woronowicz's theory are made.

Quantum supergroups of GL(n|m) type: Differential forms, Koszul complexes, and Berezinians