Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theory is the relationship between the p -local structure and the p -modular representation theory of a given finite group. In [ 5 ], Broue poses some startling conjectures. For example, he conjectures that if e is a p -block of a finite group G with abelian defect group D and if f is the Brauer correspondent block of e of the normalizer, N G ( D ), of D then e and f have equivalent derived categories over a complete discrete valuation ring with residue field of characteristic p . Some evidence for this conjecture has been obtained using an important Morita analog for derived categories of Rickard [ 11 ]. This result states that the existence of a tilting complex is a necessary and sufficient condition for the equivalence of two derived categories. In [ 5 ], Broue also defines an equivalence on the character level between p -blocks e and f of finite groups G and H that he calls a ‘perfect isometry’ and he demonstrates that it is a consequence of a derived category equivalence between e and f . In [ 5 ], Broue also poses a corresponding perfect isometry conjecture between a p -block e of a finite group G with an abelian defect group D and its Brauer correspondent p -block f of N G ( D ) and presents several examples of this phenomena. Subsequent research has provided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p -blocks e , f of finite groups G , H there are also perfect isometries between p -blocks of p -local subgroups corresponding to e and f and these isometries are compatible in a precise sense. In [ 5 ], Broue calls such a family of compatible perfect isometries an ‘isotypy’. In [ 11 ], Rickard addresses the analogous question of defining a p -locally compatible family of derived equivalences. In this important paper, he defines a ‘splendid tilting complex’ for p -blocks e and f of finite groups G and H with a common p -subgroup P . Then he demonstrates that if X is such a splendid tilting complex, if P is a Sylow p -subgroup of G and H and if G and H have the same ‘ p - local structure’, then p -local splendid tilting complexes are obtained from X via the Brauer functor and ‘lifting’. Consequently, in this situation, we obtain an isotypy when e and f are the principal blocks of G and H . Linckelmann [ 9 ] and Puig [ 10 ] have also obtained important results in this area. In this paper, we refine the methods and program of [ 11 ] to obtain variants of some of the results of [ 11 ] that have wider applicability. Indeed, suppose that the blocks e and f of G and H have a common defect group D . Suppose also that X is a splendid tilting complex for e and f and that the p -local structure of (say) H with respect to D is contained in that of G , then the Brauer functor, lifting and ‘cutting’ by block indempotents applied to X yield local block tilting complexes and consequently an isotypy on the character level. Since the p -local structure containment hypothesis is satisfied, for example, when H is a subgroup of G (as is the case in Broue's conjectures) our results extend the applicability of these ideas and methods.