Some examples on duality

An immediate corollary is that it is always possible to pick a Z-lattice inside QS so that the corresponding p-modular reduction is isomorphic to the dual of the conventional p-modular reduction (coming from the standard basis and written FpS λ). Thus our theorem gives a rich source of examples of non-isomorphic p-modular reductions. It would be interesting to know how far it is possible, by picking a suitable basis of QS λ, to make the p-modular reduction semisimple. The theorem is not hard to prove, the main idea being the following trivial generalisation of James’ Submodule Theorem.