Definition and properties of $\mathcal{B}^\dagger$

Question

I am currently trying to read Colmez' "Série principale unitaire pour $Gl_2(\mathbb{Q}_p)$ et représentations triangulines de dimension 2", that you can find here https://webusers.imj-prg.fr/~pierre.colmez/triangulines . In the proof of Lemma 4.1. at the very end I can not follow anymore.

The statement is the following: For $b\in \mathcal{E}^{\dagger}$ (i.e. an element of the Robba ring that is bounded at 0) apparently can find an element $c$ of $\mathcal{B}^\dagger$ with $\varphi(c)=bc$, where $\varphi$ is the operator with $T\mapsto (1+T)^p-1$. Furthermore there is stated that $\mathcal{B}^\dagger$ is absolutely non-ramified over $\mathcal{E}^\dagger$. In the same paper I can not find a definition of $\mathcal{B}^\dagger$, but in another paper of Colmez, "Représentations cristallines et représentation de hauteur finie", that you can find here https://webusers.imj-prg.fr/~pierre.colmez/hauteurfinie.pdf it is defined. I am still quite new to the theory of $(\varphi,\Gamma)$-modules and did not work with Witt-vectors yet. So I have truble understanding what exactly $\mathcal{B}^\dagger$ is and how those statements would follow.

I am looking for a reference where these two statements are proven. It feels like they follow fast from the definition, so I would also be happy about some reference where $\mathcal{B}^\dagger$ is regarded in greater detail.

Answer 1

The statement "if $b \in \mathcal{E}^\dagger$ then there exists $c \in \mathbf{B}^\dagger$ such that $\varphi(c)=bc$" is clearly incorrect (just take $b=p$).

The paper that you are reading is an abandoned preliminary version of other papers that were subsequently published. If you look in Colmez' corresponding published paper (the one on trianguline representations) you'll see that the proof of prop 4.2 in the paper you're reading is done quite differently (see prop 3.1 of the paper on trianguline representations).

You can find general statements about $\tilde{\mathbf{B}}^\dagger$ in other papers of Colmez, or of mine, or of Kedlaya's, for example, bearing in mind that the notation is usually different (see for instance the table on page 98 of my paper "Construction de (phi,Gamma)-modules: représentations p-adiques et B-paires"). While I'm advertising my own stuff: you can also take a look at chapter 21 of my course notes http://perso.ens-lyon.fr/laurent.berger/autrestextes/CoursIHP2010.pdf