On non-commutative Euler systems, I: preliminaries on `det' and `Fit'

We extend some classical constructions in commutative algebra to the setting of modules over orders in (non-commutative) semisimple algebras. Our theory incorporates, inter alia, `reduced' versions of the notions of higher Fitting invariants and higher exterior powers and of the Grothendieck-Knudsen-Mumford determinant functor on perfect complexes. In a companion article, these results are used to develop a theory of non-commutative Euler systems for $p$-adic representations.