### Abstract

Broue's abelian defect conjecture [Asterisque 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. Broue, 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, Broue's abelian defect group conjecture for the Held group and the sporadic Suzuki group, J. Algebra 279 (2004)638-666] to show that Broue's abelian defect group conjecture holds for nonprincipal blocks of the simple Held group and the sporadic Suzuki group. (c) 2004 Elsevier Inc. All rights reserved.

Original language | English |
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Pages (from-to) | 726-729 |

Number of pages | 3 |

Journal | Journal of Algebra |

Volume | 285 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 |

### Keywords

- Broue's conjecture
- Brauer construction
- block
- Brauer pair

## Cite this

*Journal of Algebra*,

*285*(2), 726-729. https://doi.org/10.1016/j.jalgebra.2004.08.031