Cohomological reduction by split pairs

A new method is developed to compare cohomology in module categories of different rings. This method does in general not produce isomorphisms, but surjective (or injective) maps between extension groups of modules over the two rings involved. Applications of this method are given to abstract problems—we recover and extend results on the strong no loops conjecture—and to algebras naturally coming up in invariant theory—we relate the cohomology of Brauer algebras with that of various symmetric groups.