Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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...

share|improve this question
    
If this is true the argument of Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Lemma 9.7 should give it to you. –  user1594 Jun 5 '11 at 19:28
1  
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 '11 at 19:36
    
which, btw, is available on his website: math.jussieu.fr/~nekovar/pu/sel.ps –  Rob Harron Jun 5 '11 at 19:40
    
@Rob H.: Thanks for the reference! And yes, I did want the tensor product you suggest. –  David Hansen Jun 5 '11 at 20:03
2  
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 '11 at 20:19
show 2 more comments

1 Answer 1

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.