## On equivalences for cohomological Mackey functors

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- by Markus Linckelmann PDF
- Represent. Theory
**20**(2016), 162-171 Request permission

## 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

- S. Bouc, R. Stancu, and P. J. Webb,
*On the projective dimensions of Mackey functors*, preprint (2015), arXiv:1503.03955 - Michel Broué,
*Equivalences of blocks of group algebras*, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 1–26. MR**1308978**, DOI 10.1007/978-94-017-1556-0_{1} - Markus Linckelmann,
*The source algebras of blocks with a Klein four defect group*, J. Algebra**167**(1994), no. 3, 821–854. MR**1287072**, DOI 10.1006/jabr.1994.1214 - Markus Linckelmann,
*On splendid derived and stable equivalences between blocks of finite groups*, J. Algebra**242**(2001), no. 2, 819–843. MR**1848975**, DOI 10.1006/jabr.2001.8812 - Markus Linckelmann,
*Trivial source bimodule rings for blocks and $p$-permutation equivalences*, Trans. Amer. Math. Soc.**361**(2009), no. 3, 1279–1316. MR**2457399**, DOI 10.1090/S0002-9947-08-04577-7 - Markus Linckelmann,
*Finite generation of Hochschild cohomology of Hecke algebras of finite classical type in characteristic zero*, Bull. Lond. Math. Soc.**43**(2011), no. 5, 871–885. MR**2854558**, DOI 10.1112/blms/bdr024 - Lluís Puig,
*Pointed groups and construction of characters*, Math. Z.**176**(1981), no. 2, 265–292. MR**607966**, DOI 10.1007/BF01261873 - Lluís Puig,
*Local fusions in block source algebras*, J. Algebra**104**(1986), no. 2, 358–369. MR**866781**, DOI 10.1016/0021-8693(86)90221-8 - Lluís Puig,
*On the local structure of Morita and Rickard equivalences between Brauer blocks*, Progress in Mathematics, vol. 178, Birkhäuser Verlag, Basel, 1999. MR**1707300** - Baptiste Rognerud,
*Equivalences between blocks of cohomological Mackey algebras*, Math. Z.**280**(2015), no. 1-2, 421–449. MR**3343914**, DOI 10.1007/s00209-015-1431-x - Daisuke Tambara,
*Homological properties of the endomorphism rings of certain permutation modules*, Osaka J. Math.**26**(1989), no. 4, 807–828. MR**1040426** - Jacques Thévenaz and Peter Webb,
*The structure of Mackey functors*, Trans. Amer. Math. Soc.**347**(1995), no. 6, 1865–1961. MR**1261590**, DOI 10.1090/S0002-9947-1995-1261590-5 - Tomoyuki Yoshida,
*On $G$-functors. II. Hecke operators and $G$-functors*, J. Math. Soc. Japan**35**(1983), no. 1, 179–190. MR**679083**, DOI 10.2969/jmsj/03510179

## Additional 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