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Isométries de caractères et équivalences de Morita ou dérivées. (Isometries of characters and Morita or derived equivalences). (French) Zbl 0727.20005
Let R be a noetherian integral domain with quotient field K, and let A and B be R-orders. The main purpose of the paper under review is to show that in order to prove that certain functors induce (derived or Morita) equivalences between A and B it suffices (under suitable hypotheses) to do so after extending scalars to K. The corresponding results are then applied to blocks of finite groups, in particular to l-blocks of finite reductive groups in characteristic \(p\neq l\). The author introduces the notion of a regular l-block of such a group and shows that a regular l- block B is nilpotent with abelian defect group. It thus follows from general block theory that B is Morita equivalent to the group algebra of its defect group. The author is able to show that in the case under consideration a Morita equivalence is in fact induced by a certain l-adic cohomology group. Moreover, he conjectures that this is also true in a more general situation where B is no longer nilpotent and thus general block theory does no longer apply. He then reduces this conjecture to a problem in algebraic geometry.

MSC:
20C20 Modular representations and characters
16D90 Module categories in associative algebras
20C33 Representations of finite groups of Lie type
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
18E30 Derived categories, triangulated categories (MSC2010)
20C11 \(p\)-adic representations of finite groups
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