A generalization of dual symmetry and reciprocity for symmetric algebras

We generalize these results to an arbitrary finite dimensional algebra A. Our main theorem, which is the same as the Landrock lemma for finite dimensional symmetric algebras, relates radical layers of projective indecomposable modules P to radical layers of the A-duals of those modules and to socle layers of injective indecomposable modules νP where ν is the Nakayama functor. A key tool to prove the main theorem is a pair of adjoint functors, which we call socle functors and capital functors.