Timeline for Why is the norm map dual to restriction under Tate local duality?
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10 events
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| Jul 29, 2014 at 4:04 | comment | added | user27920 | @QuestionMark: Another striking example (again expressing the initially disorienting fact that $f_{\ast}(\mu_p) = {\rm{R}}_{k'/k}(\mu_p)$ has dimension $p-1>0$ in the preceding comment), with $k'/k$ as in the preceding comment, is that applying ${\rm{R}}_{k'/k}$ to the central $k'$-isogeny ${\rm{SL}}_p \rightarrow {\rm{PGL}}_p$ yields a non-surjective $k$-homomorphism with normal image having commutative cokernel representing the fppf higher direct image ${\rm{R}}^1 f_{\ast}(\mu_p)$. | |
| Jul 28, 2014 at 16:44 | comment | added | user27920 | @QuestionMark: Sure, just let $k'/k$ be a purely inseparable extension of degree $p$ in characteristic $p$, and apply ${\rm{R}}_{k'/k}$ to the $p$-torsion Kummer sequence over $k'$. Letting $f:{\rm{Spec}}(k') \rightarrow {\rm{Spec}}(k)$ be the finite flat structure map, you get that $f_{\ast}(\mathbf{G}_m)$ is "${k'}^{\times}$ as a $k$-group", and in particular $p$-power on this $p$-dimensional smooth commutative affine group is valued in its 1-dimensional maximal $k$-torus (modulo which it becomes $p$-torsion, hence unipotent). In particular, $f_{\ast}$ applied to $p$-power isn't onto. | |
| Jul 28, 2014 at 14:00 | comment | added | Chris Wuthrich | Maybe you want to edit the question. It seems to me that the question you wanted to ask could be made more precise. | |
| Jul 28, 2014 at 13:50 | comment | added | Question Mark | @user52824: do you have an example of a nonexact finite flat pushforward? | |
| Jul 28, 2014 at 13:45 | comment | added | Question Mark | As far as I know, Milne's book doesn't discuss the comparison of pairings: it proves the duality in characteristic $0$ using one definition of the pairing and in characteristic $p$ using a completely different definition (based on biextensions, which is the correct definition, i.e., works regardless of the characteristic). I don't know of a reference for compatibility between the two pairings. The Fisher reference is written only for elliptic curves and uses the definition of the pairing that does not seem suitable in characteristic $p$; I therefore doubt that it resolves the question at hand. | |
| Jul 28, 2014 at 13:29 | comment | added | Chris Wuthrich | I wouldn't know a good reference for the comparison with the bi-extension pairing. Maybe Milne's Airhtmetic Dualities ? | |
| Jul 28, 2014 at 13:27 | comment | added | Chris Wuthrich | It was not clear to me what definition of the pairing you are using. I thought you refer to the original definition by Tate (which is away from the characteristic). The compatibility between the two pairings is given in Proposition 2.1 in dpmms.cam.ac.uk/~taf1000/papers/ctpair.html by Tom Fisher (and probably in many other places). | |
| Jul 28, 2014 at 13:16 | comment | added | Question Mark | Thanks for your response, Chris! You base your argument on the compatibility that I have alluded to in my previous comment: it would be helpful if you could indicate how to prove this compatibility (I am aware that people take this for granted)? Note that in characteristic $p$ the "biextension definition" of the pairing has to be used (in place of augmented cup products), so checking the compatibility involves dealing with edge maps in some spectral sequences, which is what makes this checking difficult for me. Any comments you may have on this would be most helpful. | |
| Jul 28, 2014 at 12:39 | comment | added | user27920 | The above before the mention of corestriction works as written even if $m > 0$ is divisible by the characteristic (using fppf cohomology and fppf Kummer sequence), so it seems the merit of Galois cohomology (which needs $m$ not divisible by the characteristic) vs. fppf cohomology is the availability of corestriction (which ultimately expresses exactness of finite etale pushforward for the etale topology, whereas finite flat pushforward is not fppf exact) and "norm". | |
| Jul 28, 2014 at 11:06 | history | answered | Chris Wuthrich | CC BY-SA 3.0 |