## On equivalences for cohomological Mackey functors

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- by Markus Linckelmann
- Represent. Theory
**20**(2016), 162-171 - DOI: https://doi.org/10.1090/ert/482
- Published electronically: May 24, 2016
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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.## References

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## Bibliographic Information

**Markus Linckelmann**- Affiliation: City University London, Department of Mathematics, London EC1V OHB
- MR Author ID: 240411
- Email: Markus.Linckelmann.1@city.ac.uk
- Received by editor(s): October 6, 2015
- Received by editor(s) in revised form: April 13, 2016
- Published electronically: May 24, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Represent. Theory
**20**(2016), 162-171 - MSC (2010): Primary 20J05
- DOI: https://doi.org/10.1090/ert/482
- MathSciNet review: 3503952