Self-equivalences of Stable Module Categories

Let P be an abelian p-group, E a cyclic p′-group acting freely on P and k an algebraically closed field of characteristic p > 0. In this work, we prove that every self-equivalence of the stable module category of k[P oE] comes from a self-equivalence of the derived category of k[P o E]. Work of Puig and Rickard allows us to deduce that if a block B with defect group P and inertial quotient E is Rickard equivalent to k[P o E], then they are splendidly Rickard equivalent. That is, Broué’s original conjecture implies Rickard’s refinement of the conjecture in this case. All of this follows from a general result concerning the self-equivalences of the thick subcategory generated by the trivial module.