Suppose $K$ is a finite extension of $\mathbf{Q}_p$, $A=K[[T_1,\dots,T_n]]$, $V$ a finiterank free $A$module, and $\rho:G_{\mathbf{Q}} \to \mathrm{GL}(V)$ a continuous Galois representation. Are there analogues of PoitouTate 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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See the recent work of Meng Fa Lim (with Sharifi) (1) PoitouTate duality over extensions of global fields (2) Nekovar duality over padic Lie extensions of global fields 

