Abstract
We show that the Green correspondence induces an injective group homomorphism from the linear source Picard group L ( B ) of a block B of a finite group algebra to the linear source Picard group L ( C ) , where C is the Brauer correspondent of B . This homomorphism maps the trivial source Picard group T ( B ) to the trivial source Picard group T ( C ) . We show further that the endopermutation source Picard group E ( B ) is bounded in terms of the defect groups of B and that when B has a normal defect group E ( B ) = L ( B ) . Finally we prove that the rank of any invertible B -bimodule is bounded by that of B .
Highlights
Let p be a prime and k a perfect field of characteristic p
An A-A-bimodule M is called invertible if M is finitely generated projective as a left A-module, as a right A-module, and if there exits an A-A-bimodule N which is finitely generated projective as a left and right A-module such that M ⊗A N ∼= A ∼= N ⊗A M as A-A-bimodules
We show that there is a local bound, without any assumption on the characteristic of O, for the order of the subgroup E(B) of isomorphism classes of invertible B-B-bimodules X having an endopermutation module as a source, for some vertex
Summary
Let p be a prime and k a perfect field of characteristic p. The corresponding invertible C-C-bimodule Y is the Green correspondent of X, and the map X → Y induces the group homomorphism L(B) → L(C) as stated in the theorem. This third step proceeds in two stages - first for the subgroup T (B), and for L(B). We show that there is a local bound, without any assumption on the characteristic of O, for the order of the subgroup E(B) of isomorphism classes of invertible B-B-bimodules X having an endopermutation module as a source, for some vertex.
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