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Alex B.'s user avatar
Alex B.'s user avatar
Alex B.'s user avatar
Alex B.
  • Member for 14 years, 3 months
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A question on algebraic independence
Straightened out the logic of the question
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If H is a quotient of G, does there exist an H-extension of \mathbb{Q} not contained in a G-extension?
Re \mathbb{Z}/p^2\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}: the same argument shows that if any \ell as in your example ramifies in the H-extension k, then k is not embeddable in a G-extension, since the local extension would have to be totally (tamely) ramified, which the hypothesis on \ell precludes. My question is: what is the smallest examples of a non-split surjective map G\to H that splits upon restriction to every soluble subgroup of G? For such a map, there won't be a local obstruction as above.
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Proper action on product manifold
To add to Mark's comment: for example if M=N, then \mathbb{R}^n\wr (\mathbb{Z}/2\mathbb{Z})=(\mathbb{R}^n\times \mathbb{R}^n)\rtimes \mathbb{Z}/2\mathbb{Z} acts properly on N\times N.
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Reference request. Finiteness of the Selmer group
Sorry about the rushed comment! I am writing from memory. I can check this myself later and may leave a proper answer.
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Reference request. Finiteness of the Selmer group
Have you looked at this paper of Kestutis Cesnavicius: imo.universite-paris-saclay.fr/~kestutis.cesnavicius/…? I think one ought to be able to adapt the usual char 0 proof, which involves the fact that the group of cohomology classes that are unramified outside a finite set of primes is finite, to the positive char case by working with fppf cohomology. The main theorem in the linked paper has some extra assumptions, but I believe that the finiteness should be extractable in full generality. Have a look, in particular, at Section 5.
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Wedderburn decomposition of special linear groups
Added details on how to compute the Artin-Wedderburn component from the relevant data
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