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 Mosc. Math. J., 2011, Volume 11, Number 4, Pages 723–803 (Mi mmj440)

Derived Mackey functors

D. Kaledinab

a Korean Institute for Advanced Studies, Seoul, Rep. of Korea
b Steklov Math. Institute, Moscow, USSR

Abstract: For a finite group $G$, the so-called $G$-Mackey functors form an abelian category $\mathcal M(G)$ that has many applications in the study of $G$-equivariant stable homotopy. One would expect that the derived category $\mathcal D(\mathcal M(G))$ would be similarly important as the “homological” counterpart of the $G$-equivariant stable homotopy category. It turns out that this is not so – $\mathcal D(\mathcal M(G))$ is pathological in many respects. We propose and study a replacement for $\mathcal D(\mathcal M(G))$, a certain triangulated category $\mathcal{DM}(G)$ of “derived Mackey functors” that contains $\mathcal M(G)$ but is different from $\mathcal D(\mathcal M(G))$. We show that standard features of the $G$-equivariant stable homotopy category such as the fixed points functors of two types have exact analogs for the category $\mathcal{DM}(G)$.

Key words and phrases: derived, Mackey functor.

DOI: https://doi.org/10.17323/1609-4514-2011-11-4-723-803

Full text: http://www.ams.org/.../abst11-4-2011.html
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MSC: 18G99
Received: December 15, 2008; in revised form August 16, 2010
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Citation: D. Kaledin, “Derived Mackey functors”, Mosc. Math. J., 11:4 (2011), 723–803

Citation in format AMSBIB
\Bibitem{Kal11} \by D.~Kaledin \paper Derived Mackey functors \jour Mosc. Math.~J. \yr 2011 \vol 11 \issue 4 \pages 723--803 \mathnet{http://mi.mathnet.ru/mmj440} \crossref{https://doi.org/10.17323/1609-4514-2011-11-4-723-803} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2918295} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000300368300004} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. B. Kaledin, “Cyclotomic complexes”, Izv. Math., 77:5 (2013), 855–916
2. Barwick C., “Spectral Mackey Functors and Equivariant Algebraic K-Theory (i)”, Adv. Math., 304 (2017), 646–727
3. Positselski L., “Dedualizing Complexes of Bicomodules and Mgm Duality Over Coalgebras”, Algebr. Represent. Theory, 21:4 (2018), 737–767
4. Berman J.D., “On the Commutative Algebra of Categories”, Algebr. Geom. Topol., 18:5 (2018), 2963–3012