Suppose $K$ is a finite extension of $\mathbf{Q}_p$, $A=K[[T_1,\dots,T_n]]$, $V$ a finite-rank free $A$-module, and $\rho:G_{\mathbf{Q}} \to \mathrm{GL}(V)$ a continuous Galois representation. Are there analogues of Poitou-Tate dualtiy and the Tate local and global Euler characteristic formulas for the cohomology groups $H^i_{\mathrm{cont}}(G_{\mathbf{Q}},V)$, $H^i_{\mathrm{cont}}(G_{\mathbf{Q}_{\ell}},V)$? I was hoping I could "take the limit" of the formulas for the groups $H^i(G,V/(T_1^j,\dots,T_n^j)V)$ as $j\to\infty$, but that doesn't seem so easy...
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asked Jun 5, 2011 at 19:08
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$\begingroup$ If this is true the argument of Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Lemma 9.7 should give it to you. $\endgroup$– user1594Commented Jun 5, 2011 at 19:28
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1$\begingroup$ You could take a look Nekovář's Selmer complexes. Also, you probably want $\mathcal{O}_K[[T_1,\dots,T_n]]$ (or that tenor $K$). $\endgroup$– Rob HarronCommented Jun 5, 2011 at 19:36
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$\begingroup$ which, btw, is available on his website: math.jussieu.fr/~nekovar/pu/sel.ps $\endgroup$– Rob HarronCommented Jun 5, 2011 at 19:40
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$\begingroup$ @Rob H.: Thanks for the reference! And yes, I did want the tensor product you suggest. $\endgroup$– David HansenCommented Jun 5, 2011 at 20:03
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2$\begingroup$ Have you asked Jay Pottharst or Joel Bellaiche, both of whom are in your neighbourhood, and both of whom will have thought about this kind of question? Regards, Matthew $\endgroup$– EmertonCommented Jun 5, 2011 at 20:19
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1 Answer
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See the recent work of Meng Fa Lim (with Sharifi) (1) Poitou-Tate duality over extensions of global fields
(2) Nekovar duality over p-adic Lie extensions of global fields
