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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?

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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
@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
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".

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