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I apologize in advance is this is not the correct “category” for this discussion. Please feel free to fix this.
Under the realization functors subsection of the entry on motivic homotopy theory, the last line reads:
“For a non-separably closed field k, there is a Gal(k^sep/k)-equivariant realization analogous to the Real realization.”
However no reference is given, and I do not believe such a reference exists. This is an extremely subtle point. See for example Wickelgren’s paper:
http://people.math.gatech.edu/~kwickelgren3/papers/Etale_realization.pdf
This does work in the unstable setting, but this is currently being written up by myself and Elden Elemanto, and we are working on the stable result. I’m not sure the best way to revise this entry. But for sure, if there is a reference, it should be added, and otherwise, I would suggest removing or rephrasing this sentence.
Thanks for the alert. This statement was added in rev 17 by Marc Hoyois. I suppose there is a good chance that he will see this discussion here, otherwise you might contact him by email.
I only just found this thread. This was indeed inaccurate, so I removed it. There is only a non-genuine Galois-equivariant realization, and the target is a nonstandard category of pro-spectra where not only S^1 but also its Tate twist (= the etale homotopy type of $\mathbf G_m$) are inverted. I believe that a genuine Galois-equivariant realization exists iff $k$ is a real closed field. There are no references for any of these claims, however.
Hi Marc, whilst you’re here, I don’t suppose you’d have any thoughts about my question here? (And feel free to improve on what I wrote at Tate twist :-))
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