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…
$$ ( E_{hC_p} \to E^{C_p} \to E^{\Phi C_p} ) \to ( E_{hC_p} \to E^{hC_p} \to E^{tC_p}) . $$
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…
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 $\Phi^{C_p}E=E^{tC_p}=0$
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.)
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…
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