The indecomposability of a certain bimodule given by the Brauer construction

Koshitani, S. & Linckelmann, M. (2005). The indecomposability of a certain bimodule given by the Brauer construction. Journal of Algebra, 285(2), pp. 726-729. doi: 10.1016/j.jalgebra.2004.08.031

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Abstract

Broué’s abelian defect conjecture [3, 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 [11], 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. [4, 6.3], [8, 3.1], [12, 4.1], and [13, 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 [7] to show that Broué’s abelian defect group conjecture holds for nonprincipal blocks of the simple Held group and the sporadic Suzuki group.

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: School of Engineering & Mathematical Sciences > Department of Mathematical Science
URI: http://openaccess.city.ac.uk/id/eprint/1905

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