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On equivalences for cohomological Mackey functors


Author: Markus Linckelmann
Journal: Represent. Theory 20 (2016), 162-171
MSC (2010): Primary 20J05
DOI: https://doi.org/10.1090/ert/482
Published electronically: May 24, 2016
MathSciNet review: 3503952
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Abstract: By results of Rognerud, a source algebra equivalence between two $ p$-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above mentioned derived equivalence. We show further that a splendid stable equivalence of Morita type between two blocks induces an equivalence between the categories of cohomological Mackey functors which vanish at the trivial group. We observe that the module categories of a block, the category of cohomological Mackey functors, and the category of cohomological Mackey functors which vanish at the trivial group arise in an idempotent recollement. Finally, we extend a result of Tambara on the finitistic dimension of cohomological Mackey functors to blocks.


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Additional Information

Markus Linckelmann
Affiliation: City University London, Department of Mathematics, London EC1V OHB
Email: Markus.Linckelmann.1@city.ac.uk

DOI: https://doi.org/10.1090/ert/482
Received by editor(s): October 6, 2015
Received by editor(s) in revised form: April 13, 2016
Published electronically: May 24, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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