The indecomposability of a certain bimodule given by the Brauer construction

Abstract

Broué's abelian defect conjecture [Astérisque 181/182 (1990) 61–92, 6.2] predicts for a p-block of a finite group G with an abelian defect group P a derived equivalence between the block algebra and its Brauer correspondent. By a result of Rickard [J. London Math. Soc. 43 (1991) 37–48], such a derived equivalence would in particular imply a stable equivalence induced by tensoring with a suitable bimodule—and it appears that these stable equivalences in turn tend to be obtained by “gluing” together Morita equivalences at the local levels of the considered blocks; see, e.g., [M. Broué, Equivalences of blocks of group algebras, in: V. Dlab, L.L. Scott (Eds.), Finite Dimensional Algebras and Related Topics, Kluwer Acad. Publ., 1994, pp. 1–26, 6.3], [M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, J. Algebra 242 (2001) 819–843, 3.1], [J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996) 331–358, 4.1], and [R. Rouquier, Block theory via stable and Rickard equivalences, in: M.J. Collins, B.J. Parshall, L.L. Scott (Eds.), Modular Representation Theory of Finite Groups, de Gruyter, Berlin, 2001, pp. 101–146, 5.6, A.4.1]. This note provides a technical indecomposability result which is intended to verify in suitable circumstances the hypotheses that are necessary to apply gluing results as mentioned above. This is used in [S. Koshitani, N. Kunugi, K. Waki, Broué's abelian defect group conjecture for the Held group and the sporadic Suzuki group, J. Algebra 279 (2004)638–666] to show that Broué's abelian defect group conjecture holds for nonprincipal blocks of the simple Held group and the sporadic Suzuki group.