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I'm looking for a reference for the decomposition of the category of Mackey functors for a finite group when the order of the group is inverted. (There is also an analogous decomposition for the category of $G$-spectra.)

Let $\mathrm{Mack}_G$ denote the category of Mackey functors for a finite group $G$. Given a subgroup $H \leq G$, we let $W_H$ denote the Weyl group of $H$. For each $H$, we have a functor from $\mathrm{Mack}_G \to \mathrm{Mod}( \mathbb{Z}[W_H])$ which sends a Mackey functor $M$ to the quotient of $M(H)$ by the transfers from proper subgroups. We get a functor

$$\mathrm{Mack}_G \to \prod_H \mathrm{Mod}( \mathbb{Z}[W_H])$$

when we take the product over a system of representatives of conjugacy classes of subgroups.

This functor is not an equivalence, but it becomes an equivalence when we invert $|G|$ - in fact, this decomposition comes from the splitting of the Burnside ring of $G$ when $|G|$ is inverted.

I'm looking for a reference for this (classical) fact. Most of the sources that come close to this only state the result rationally, but I suspect this is in the literature somewhere. Any help would be appreciated!

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  • $\begingroup$ Try looking at papers by Jacques Thevenaz. $\endgroup$
    – Geoff Robinson
    Commented Oct 25, 2016 at 12:18
  • $\begingroup$ Dear Akhil, does Yoshida's "Idempotents in Burnside rings and Dress Induction Theorem" accomplish this for you? Unfortunately it's not stated as an equivalence of categories that I can see, but basically all of the heavy lifting is done. $\endgroup$
    – Tyler Lawson
    Commented Oct 25, 2016 at 13:47
  • $\begingroup$ (I was nearly certain that this result was in Lewis' "Theory of Green functors" notes but have been unable to find it this morning.) $\endgroup$
    – Tyler Lawson
    Commented Oct 25, 2016 at 13:56
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    $\begingroup$ @TylerLawson: Thank you! Lewis's notes are available on Ravenel's website (web.math.rochester.edu/people/faculty/doug/otherpapers/…). It looks like this is discussed at the very end. $\endgroup$
    – Akhil Mathew
    Commented Oct 25, 2016 at 20:13

1 Answer 1

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In case anyone stumbles across this question, there is now a published reference in this generality: Theorem 3.4.22 of Schwede's Global Homotopy Theory book.

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