We define and study sl2 -categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Brou´’s abelian defect group conjecture for symmetric groups. e We give similar results for general linear groups over finite fields. .
