Timeline for $p$-completeness of the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\infty} BK)$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 19, 2020 at 2:45 | vote | accept | Victor TC | ||
| Sep 19, 2020 at 17:08 | |||||
| Sep 19, 2020 at 1:57 | comment | added | Tom Goodwillie | A very slightly different point of view: A spectrum $C$ is $p$-complete iff $F(A,C)=0$ for every $S/p$-acyclic spectrum $A$, i.e. every $A$ such that $A/p=0$. A spectrum $T$ is $p$-torsion if and only if $F(T,A)=0$ for every $S/p$-acyclic $A$. So if $T$ is $p$-torsion then for every $U$ the spectrum $F(T,U)$ is $p$-complete since, for every such $A$, $F(A,F(T,U))=F(T,F(A,U))=0$. The key point is that $F(A,U)$ is always $S/p$-acyclic if $A$ is $S/p$-acyclic. ($F(A,U)/p$ is a shift of $F(A/p,U)$.) | |
| Sep 18, 2020 at 23:25 | history | edited | Piotr Pstrągowski | CC BY-SA 4.0 |
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| Sep 18, 2020 at 23:23 | comment | added | Tim Campion | (Sorry I deleted my comment -- I was trying to edit it to say thanks for the addition, but went longer than 5 minutes and then ended up deleting it!) Interesting that both arguments for $T$ $p$-torsion $\Rightarrow F(T,U)$ $p$-complete seem to rely on the fact that $M(p)$ is a shift of its own Spanier-Whitehead dual -- especially in your formulation, this seems like the least "formal" fact used. | |
| Sep 18, 2020 at 23:16 | comment | added | Piotr Pstrągowski | I added an argument using Maschke's theorem. | |
| Sep 18, 2020 at 23:14 | history | edited | Piotr Pstrągowski | CC BY-SA 4.0 |
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| Sep 18, 2020 at 23:00 | history | answered | Piotr Pstrągowski | CC BY-SA 4.0 |