Equivalences of Derived Categories for Symmetric Algebras

It is about a decade since Broué made his celebrated conjecture [2] on equivalences of derived categories in block theory: that the module categories of a block algebra A of a finite group algebra and its Brauer correspondent B should have equivalent derived categories if their defect group is abelian. Since then, character-theoretic evidence for the conjecture has accumulated rapidly, but until very recently there have been very few examples where the conjecture has actually been verified. This is because the precise structure of, say, the indecomposable projective modules for A is known only in the simplest cases (although the corresponding structure for B is much easier to determine): this makes it very difficult to carry out explicit calculations to verify an equivalence of derived categories. Recently, however, Okuyama [6] introduced a method of proving that there is an equivalence of derived categories that needs very little explicit information about A. In many of the simpler cases where Broué’s conjecture is not yet known to be true, there is known to be a ‘stable equivalence of Morita type’ between A and B: an exact functor between the module categories that is an equivalence of categories ‘modulo projective modules’. This is a consequence of an equivalence of derived categories, since the stable module category is a canonical quotient of the derived category. Moreover, recent work of Rouquier [11],[12] gives a method of constructing such stable equivalences from equivalences of derived categories for smaller groups. Okuyama’s method is a strategy for lifting stable equivalences to equivalences of derived categories. If one can produce an equivalence of derived categories between B and a third algebra C, and if the objects of the derived category D(mod(B)) that correspond to the simple C-modules are isomorphic in the stable module category of B (regarded as a quotient category of D(mod(B))) to the images of the simple A-modules under a stable equivalence of Morita type, then it follows from a theorem of Linckelmann [5, Theorem 2.1] that A and C are Morita equivalent, and so A and B have equivalent derived categories. Note that to carry out this strategy, one needs to know nothing about A except which objects of the stable module category of B correspond to the simple A-modules. Okuyama used this method to verify Broué’s conjecture for many examples of blocks with defect group C3 × C3 [6]. This still leaves the problem of finding a suitable equivalence