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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.

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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