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1. • CommentRowNumber1.
   • CommentAuthorDavidCarchedi
   • CommentTimeSep 14th 2017
   • PermaLink
   Author: DavidCarchedi
   Format: MarkdownItexI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 14th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the alert. This statement was added in [rev 17](https://ncatlab.org/nlab/revision/diff/motivic+homotopy+theory/17#realization_functors) by Marc Hoyois. I suppose there is a good chance that he will see this discussion here, otherwise you might contact him by email.

   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.

3. • CommentRowNumber3.
   • CommentAuthorMarc Hoyois
   • CommentTimeMar 5th 2018
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   Author: Marc Hoyois
   Format: MarkdownItexI 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorRichard Williamson
   • CommentTimeMar 5th 2018
   • (edited Mar 5th 2018)
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   Author: Richard Williamson
   Format: MarkdownItexHi Marc, whilst you're here, I don't suppose you'd have any thoughts about my question [here](https://nforum.ncatlab.org/discussion/8278/tate-twists-and-twisted-cohomology/)? (And feel free to improve on what I wrote at [[Tate twist]] :-))

   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 :-))