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3:37 AM
i have a very basic equivariant question (or hopefully just reference request). i want to understand the isotropy separation sequence & tate gluing square for finite $C_p$-suspension spectra. for this, it suffices to understand (1) those for $C_p/e$ and $C_p/C_p$ [leaving $\Sigma^\infty_{C_p,+}$ implicit] and (2) the induced maps between them -- perhaps even just those of degree 0 in my case. so, i'd like to make sure i've got (1) right, and then see if anyone can help with (2).
for general $E$, i'll write
$$ ( E_{hC_p} \to E^{C_p} \to E^{\Phi C_p} ) \to ( E_{hC_p} \to E^{hC_p} \to E^{tC_p}) . $$
for $E = C_p/e$, i believe this is:
$$ (S \to S \to 0 ) \to ( S \to (C_p/e)^{hC_p} \to (C_p/e)^{tC_p} ).$$
(i don't know how to identify the last two terms.) for $E = C_p/C_p$, i believe this is:
$$ ( S_{hC_p} \to S_{hC_p} \oplus S \to S ) \to (S_{hC_p} \to S^{hC_p} \to S^\wedge_p ). ) $$
of course, i can also identify $S_{hC_p} \simeq \Sigma^\infty_+ BC_p$. i don't know how to identify $S^{hC_p}$ otherwisely, either; equivalently, i don't know what the gluing map $S^\wed
can anyone improve my understanding of this situation? (i may well have made some mistakes, in addition to failing to identify some of the terms.)
regarding (2), i'd like to understand in these terms both the map $\Sigma^\infty_{C_p,+}(C_p/e \to C_p/C_p)$ and its dual. and as i implied above, i'd even be interested to know how this all works after rationalization, just as a first approximation.
 
 
3 hours later…
6:39 AM
@AaronMazel-Gee $E=C_p/e$ is both Borel and coBorel (or cofree and free, if you prefer), so $E_{hC_p}=E^{hC_p}=\mathbb{S}$ and $\Phi^{C_p}E=E^{tC_p}=0$
(there is the useful observation that every time you have a commutative ring that is coBorel, then every module over it is automatically Borel - in particular the ring is Borel as well)
@AaronMazel-Gee For this I'm not sure exactly of what you are looking for, beyond the fact that this map induces the tom Dieck splitting, as you correctly noted.
 
 
2 hours later…
8:22 AM
Does anyone know a reference for p-adic Atiyah-Segal?
By this I mean that for a $p$-group $G$, the canonical map $KU_{p,G}(*) -> KU_p(BG)$ is an isomorphism.
This is well known but I coundn't find reference which proves it.
(This includes the fact that the augmentaion ideal defines the p-adic topology in this situataion, and a bit more.)
 
 
13 hours later…
9:25 PM
@ShayBenMoshe Atiyah & Tall, Part III, section 1, seems to have this.
 

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