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...

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 

