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

This question is about p-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ and $(\varphi, \Gamma)$-modules. By theorems of Fontaine, Cherbonnier-Colmez and Kedlaya, the category of p-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ is equivalent to each of the following three categories:

  • etale $(\varphi, \Gamma)$-modules over Fontaine's ring $\mathbb{B}_{\mathbb{Q}_p}$
  • etale $(\varphi, \Gamma)$-modules over the subring $\mathbb{B}^{\dagger}_{\mathbb{Q}_p}$
  • slope zero $(\varphi, \Gamma)$-modules over the Robba ring $\mathcal{R}$ (also known as $\mathbb{B}^{\dagger}_{\mathrm{rig}, \mathbb{Q}_p}$).

It's well known that slope 0 $(\varphi, \Gamma)$-modules over the Robba ring can sometimes be written as extensions of other Robba-ring $(\varphi, \Gamma)$-modules which are not themselves of slope 0. (Indeed there is the whole rich theory of trianguline representations, whose Robba-ring $(\varphi, \Gamma)$-modules are built up entirely from rank 1 pieces.)

My question: does this happen for either of the other two categories of $(\varphi, \Gamma)$-modules? Can one have a short exact sequence of $(\varphi, \Gamma)$-modules over $\mathbb{B}_{\mathbb{Q}_p}$ or $\mathbb{B}^{\dagger}_{\mathbb{Q}_p}$ where the middle term is etale but the two end terms are not?

share|improve this question
    
The last category is closed under extensions. What you mean is something different, isn't it? –  Laurent Berger Oct 24 '10 at 11:14
    
Sorry, I phrased that badly -- I will correct it. –  David Loeffler Oct 24 '10 at 11:24
    
Sorry if this is a naive question: when you say "p-adic representation" do you mean continuous with no other conditions? –  S. Carnahan Oct 24 '10 at 11:38
2  
@Scott: I'm pretty sure he does (phi-gammas see everything: see Fontaine's paper in the Grothendieck Festschrift). –  Kevin Buzzard Oct 24 '10 at 11:39

1 Answer 1

up vote 7 down vote accepted

In the first two cases, the slopes of $\varphi$-modules are given by the "standard" Dieudonné-Manin decomposition. In particular, subobjects of étale objects are étale.

For more info, see (for example) chapter 4.5 of Kedlaya's "Slope Filtrations Revisited".

share|improve this answer

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.