Abstract
We keep the notation of Section 6; as in 6.6 and 6.7, P .. is a vertex and N .. an O P .. -source of M .., set S .. = End O (N ..) and denote by σ: P .. → P and σ’: P ..; →P’; the surjective group homomorphisms determined by the projection maps on G × G’;. As we say in the introduction, in all the known situations where O has characteristic zero and M .. defines a Morita stable equivalence between b and b’, it turns out that σ and σ’ are both bijective and that P .. stabilizes by conjugation an O-basis of S ..- that is to say, S .. is a so-called Dade P ..-algebra (cf. 2.2) - which implies that p does not divide rank o (N ..) (cf. [38, (28.11)]). In this section we prove that all these statements are in fact equivalent to each other (see Corollary 7.4 below), and whenever they hold we say that the Morita stable equivalence between b and b’ defined by M .. is basic. The most difficult part of the proof, namely the existence of a P ..-stable O-basis in S .., is a consequence of the following general (and surprising!) result. Recall that a nondegenerate symmetric O-form µ: B → O of an O-free O-algebra B is an O-linear map fulfilling µ(aa’) = µ(a’a) for any a, a’ ∈ B and inducing an O-module isomorphism B ≅ Hom O (B, O) which maps a ∈ B on the O-form mapping a’ ∈ B on µ(aa’).
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© 1999 Springer Basel AG
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Carreres, L.P. (1999). Basic Morita stable equivalences between Brauer blocks. In: On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks. Progress in Mathematics, vol 178. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8693-2_7
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DOI: https://doi.org/10.1007/978-3-0348-8693-2_7
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9732-7
Online ISBN: 978-3-0348-8693-2
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