# Mackey(also Green and Tambara) functors and Greenlees-May

This is somewhat related to a question that I asked on Math.SE but, sadly, received no response. I apologize ahead of time if this is not appropriate for MO. Feel free to vote to close if this is the case.

I really have two (distinct) questions. The first is regarding a paper by Greenlees and May and the second is more of a "big-picture" question with no explicit relationship to their paper.

Let $G$ be a finite group.

In their paper Some Remarks On the Structure of Mackey functors , Greenlees and May define the functor:

$R: GMod \rightarrow M[G]$ where $GMod$ is the category of finite left $G$-modules and $M[G]$ is the category of $G$ Mackey functors by:

$RV(G/H) = V^H$ where $V$ is a $G$ module and $V^H$ is the $H$ fixed point set of $V$.

In their main theorem(Thm. 12) they consider the map $\eta: M \rightarrow RM(G/H_{j,k})$.

Question 1: In Theorem 12, why are $coker(\eta)$ and $ker(\eta)$ in $\mathcal{A}$? Unfortunately I don't see how this clearly follows from the induction hypothesis at the moment.

Question 2: In general, What are some examples of added benefits (aside from additional structure) that one obtains when it is known that you have a Green or Tambara functor rather than just a Mackey functor?

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I thank you for the careful reading and apologize for the concision. This is a downwards induction on the size of subgroups. Using the explicit description of the $RV$ given top of page 239 and the conventions recalled at the bottom of page 240, we arrived at the description of the relevant $RV$ given just below Prop. 7. The map $\eta$ is the identity on $M(G/H_{j,k})$ and since $M(G/H_{j',k'}) = 0$ for $j'< j$ and for $j'=j$ and $k'< k$, the same is true of $RM(G/H_{j,k})$. Therefore $Ker(\eta)$ and $Coker(\eta)$ can only be non-zero on $G/H_{j',k'}$ where $j'>j$ or $j'=j$ and $k'>k$ and hence they are in $\mathcal{A}$ by the induction hypothesis.
@PeterMay Thank you for your response! That makes much more sense now. Personally, the definition on the bottom of page 238 (definition 1) of $RV$ in terms of the $H-$fixed points seems more natural. Is there is a way to see $ker$ and $coker$ being in $\mathcal{A}$ without appealing to the explicit construction given on the top of page 239? My second question is not directly related to your paper. I was simply asking (in general) as to what one gains from having a Green or Tambara functor instead of a run-of-the-mill Mackey functor. I will edit as to make this distinction clear. Thanks! – confusedmath Sep 19 '12 at 2:07
That is not what one expects from analogy with simpler structures. It would be of interest is to compute the box product'' $RV\Box RW$ in terms of the additive description of Mackey functors that Greenlees and I gave. The point is that a Green functor $M$ is specified by a product $M\Box M \to M$. The calculation should not be hard in the simplest cases (cyclic groups of prime order say), and as far as I know has not been done.