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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...

flag	asked Jun 5 2011 at 19:08 David Hansen 8,429●2●18●48

If this is true the argument of Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Lemma 9.7 should give it to you. – JT Jun 5 2011 at 19:28

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You could take a look Nekovář's Selmer complexes. Also, you probably want $\mathcal{O}_K[[T_1,\dots,T_n]]$ (or that tenor $K$). – Rob Harron Jun 5 2011 at 19:36

which, btw, is available on his website: math.jussieu.fr/~nekovar/pu/sel.ps – Rob Harron Jun 5 2011 at 19:40

@Rob H.: Thanks for the reference! And yes, I did want the tensor product you suggest. – David Hansen Jun 5 2011 at 20:03

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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 – Emerton Jun 5 2011 at 20:19

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SGP · Answer 1 · 2011-06-06 11:14:39Z UTC

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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

answered Jun 6 2011 at 11:14

SGP
2,614●4●14

Poitou-Tate dualities for Galois representations into power series rings?

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