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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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  • $\begingroup$ If this is true the argument of Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Lemma 9.7 should give it to you. $\endgroup$
    – user1594
    Jun 5, 2011 at 19:28
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    $\begingroup$ 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 Harron
    Jun 5, 2011 at 19:36
  • $\begingroup$ which, btw, is available on his website: math.jussieu.fr/~nekovar/pu/sel.ps $\endgroup$
    – Rob Harron
    Jun 5, 2011 at 19:40
  • $\begingroup$ @Rob H.: Thanks for the reference! And yes, I did want the tensor product you suggest. $\endgroup$ Jun 5, 2011 at 20:03
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    $\begingroup$ 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 $\endgroup$
    – Emerton
    Jun 5, 2011 at 20:19

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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

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