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We show that the splendid Rickard complexes for blocks with Klein four defectgroups constructed by Rickard and Linckelmann descend to non-split fields. As acorollary, Navarro's refinement of the Alperin-McKay conjecture holds forblocks with a Klein four defect group. We also prove that splendid Moritaequivalences between blocks and their Brauer correspondents (if exist) descendto non-split situations.

Descent of equivalences for blocks with Klein four defect groups