Computations in the derived module category.

This thesis is centred around computations in the derived module category of finitely generated lattices over the integral group ring of a finite group G. Building upon the representability of the cohomology functor in the derived module category in dimensions greater than 0, we give a new characterisation of the cohomology of lattices in terms of their G-invariants, only having the syzygies of the trivial lattice to keep track of dimension. With the example of the dihedral group of order 6 we show that this characterisation significantly simplifies computations in cohomology. In particular, we determine the Bieberbach groups, that is, the fundamental groups of compact at Riemannian manifolds, with dihedral holonomy group of order 6. Furthermore, we give an interpretation of the cup product in the derived module category and show that it arises naturally as the composition of morphisms. Inspired by the graded-commutativity of the cup product in singular cohomology we give a sufficient condition for the cohomology ring of a lattice to be graded-commutative in dimensions greater than 0.