The Bernstein-Gelfand-Ponomarev Reflection Functors

S simple A-module, projective, inj.dim. S ≤ 1.
Let G   be the full subcategory of mod A of the modules whithout a direct summand S.

Then there is an algebra B and a simple injective B-module S' with proj.dim. S' ≥ 1,
    such that the full subcategory Y of the B-modules whithout a direct summand S'
    is equivalent to G.

If A is given by a quiver with relations,
then S corresponds to a sink s with no relation ending in it.
B is obtained by just reversing all the arrows ending in s ("reflection").
 
The corresponding module categories:


More generally, assume that S is a projective simple A-module.
There is a "tilting" module T (introduced by Auslander-Platzek-Reiten),
and a corresponding tilting functor HomA(T,-) : mod A → mod B, with B = End(T).