Structure theorem for Iwasawa modules over $p$-adic rings of integers

Question

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.

Answer 1

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).

Answer 2

For the non-commutative case, whilst the result is not explicitly there, almost all the work is done in 'Modules over Iwasawa algebras' by Coates, Schneider and Sujatha (see https://ivv5hpp.uni-muenster.de/u/pschnei/publ/pap/micro3.pdf ). Though the main theorem is only stated in the case where the coefficients are $\mathbb{Z}_p$, the only step where this is used is in the statement and proof of Proposition 6.2.

To see that the condition that the coefficients are $\mathbb{Z}_p$ rather than the more general $\mathcal{O}_K$ as in your question is not used in any serious way you can look at Corollary 28.6 of Schneider's book '$p$-adic Lie groups' together with Lemma 6.3 of 'Modules over Iwasawa algebras'.