Q:

A question on finding an inverse image of $\varphi-1$ over some $(\varphi, \Gamma)$-module, from an article by Cherbonnier and Colmez.

I am reading the article THEORIE D’IWASAWA DES REPRESENTATIONS p-ADIQUES D’UN
CORPS LOCAL by Cherbonnier and Colmez
https://webusers.imj-prg.fr/~pierre.colmez/CCjams.pdf
Question:
I have a question on Proposition I.4.1 (i) of the above article:  How can I find the $b$ claimed? More precisely,  for any given $(x, y)\in D(V)\oplus D(V)$ such that $(\gamma-1)x=(\varphi-1)y$, how to find $b\in D(V)$ that satisfies $(\varphi-1)b=x$? 
Remark:
As far as I know, $\varphi-1$ has inverse map only for some open subset of $D(V)$. Refer to: http://www.numdam.org/article/BSMF_1998__126_4_563_0.pdf
and is surjective on some special representations, for example, $Ind_{G_K}(V)$, the induced module of $V$. Refer to: https://tel.archives-ouvertes.fr/tel-00379771/document

A:

As pointed by an expert, $(\varphi-1)b=x$ is not solved by some $b\in D(V)$. Actually it is solved, as claimed in the article, by some element $b\in A\otimes_{\mathbb{Z}_p} V$. Indeed,  $\varphi-1$ is surjective on $A$ and hence surjective on $A\otimes_{\mathbb{Z}_p} V$, since there is no $\varphi$-action on the representation $V$ ($\varphi$ acts only on the coefficient). 
To say more for the map $\varphi-1$ on $D(V)$. Consider the following complex:
$$ 0\to D(V) \xrightarrow{\varphi-1} D(V) \to 0 $$
If $D(V):=(A\otimes V)^{G_L}$, where $G_L$ is the absolute Galois group of some precised Galois extension $L/K$. Then the cohomology of this complex computes $H^i(G_L, V)$. 
More precisely, one can easily see that the cohomology of the complex gives an $\delta$-functor: { $F^i$}: $Rep_{\mathbb{Z}_p}(G_K)\to \mathbb{Z}$-module. $F^0(V)$ coincides with $H^0(G_L, V)$  and  { $F^i$} is effaceable. The effaceability can be shown as in Ribeiro's thesis by using the embedding  of $V$ into its induced module $Ind_{G_K}V$. Indeed, one has $H^1(G_L, Ind_{G_K}V)=0$. 
Hence $H^1(G_L, V)$ vanish or not tells whether we have surjectivity for $\varphi-1$, which we know is not true in general.

