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

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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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2 Answers

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.

For the second question, I see no obvious relationship between our additive description of Mackey functors and any multiplicative structure that might be present. Anything anyone can say about that would be welcome. I've not thought hard about the question.

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@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 at 2:07
@PeterMay  Also, I was wondering if there are analogous theorems for Green/Tambara functors.  Are all Green (resp.Tambara) functors built from "simpler" Green (resp. Tambara) functors? – confusedmath Sep 19 at 2:16

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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.

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