Let $K/\mathbb Q\_p$ be a finite extension, and $\mathcal O\_K$ the ring of integers of $K$. I am asking for a reference for a structure theorem
of finitely generated modules over the completed group algebra $\mathcal O\_K[[\mathbb Z\_p]]$.


I am familiar with structure theorems for $\Lambda = \mathbb Z\_p[[\mathbb Z\_p]]$, and
more generally for $\mathbb Z\_p[[G]]$ where $G$ is a $p$-adic Lie group. I believe that analogous structure theorems should hold in the $\mathcal O\_L$ coefficent case, but I was unable to find an explicit reference.


Motivation: rings of the form $\mathcal O\_K[[\mathbb Z\_p]]$ arise naturally in the
$p$-adic Langlands program.



Structure theorem for Iwasawa modules over $p$-adic rings of integers
Serre, Jean-Pierre, *Classes des corps cyclotomiques* (Séminaire Bourbaki décembre 1958) lemme 5 p. 90 contains a statement and proof of a structure theorem (up to pseudo-isomorphism) for finitely generated modules over a noetherian integrally closed domain $A$, of which $\mathcal O\_{K}[[\mathbb Z\_{p}]]$ is a very particular example. Not that the proof in that general setting is no harder than in the case of $A=\mathbb Z\_{p}[[X]]$.


Applied to a regular local ring of dimension 2, this lemma translates in what you want (provided you know that a reflexive module over such a ring is free, which is also proved in the same reference, lemme 6 p. 91).


