A remark on an integral structure of the imperfect coefficient ring of $(\phi ,\mathrm{\Gamma })$-modules

Let $K$ be a complete discrete valuation field of characteristic $0$ with perfect residue field of characteristic $p>0$. Let ${\mathbb{A}}_K$ denote the imperfect coefficient ring of $(\phi ,\mathrm{\Gamma })$-modules defined by Jean-Marc Fontaine. We prove that the canonical map $W(k_{K_{\mathrm{\infty }}})[[\mu ]]\to {\mathbb{A}}_K\cap A_{\mathrm{i}\mathrm{n}\mathrm{f}}$ is an isomorphism, even when $K$ is ramified. This fact was remarked by Nathalie Wach without proof. In Appendix 2, we include a result of Dylan Pentland. Both results indicate the difficulty of constructing a coefficient ring of ``Wach modules’’ in the ramified case.