Multivariable period rings of $p$-adic false Tate curve extension

Let $p\ge 3$ be a prime number and $K$ be a finite extension of ${\mathbf{Q}}_p$ with uniformizer ${\pi }_K$. In this article, we introduce two multivariable period rings ${\mathbf{A}}_{\mathfrak{F},K}^{\mathrm{np}}$ and ${\mathbf{A}}_{\mathfrak{F},K}^{\mathrm{np},\mathrm{c}}$ for the étale $(\phi ,{\mathrm{\Gamma }}_{\mathfrak{F},K})$-modules of $p$-adic false Tate curve extension $K({\pi }_K^{1\mathrm{/}{p}^{\mathrm{\infty }}},{\zeta }_{p^{\mathrm{\infty }}})$. Various properties of these rings are studied and as applications, we show that $(\phi ,{\mathrm{\Gamma }}_{\mathfrak{F},K})$-modules over these rings bridge $(\phi ,\mathrm{\Gamma })$-modules and $(\phi ,\tau )$-modules over imperfect period rings in both classical and cohomological sense, which answers a question of Caruso. Finally, we construct the $\psi$ operator for false Tate curve extension and discuss the possibility to calculate Iwasawa cohomology for this extension via $(\phi ,{\mathrm{\Gamma }}_{\mathfrak{F},K})$-modules over these rings.