A generalization of dual symmetry and reciprocity for symmetric algebras

Slicing a module into semisimple ones is useful to study modules. Loewy structures provide a means of doing so. To establish the Loewy structures of projective modules over a finite dimensional symmetric algebra over a field $F$, the Landrock lemma is a primary tool. The lemma and its corollary relate radical layers of projective indecomposable modules to radical layers of the $F$-duals of those modules (“dual symmetry”) and to socle layers of those modules (“reciprocity”). 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 $\nu P$ where $\nu (-)$ 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.