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

Let $p\geq 3$ 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 $\mathbf{A}_{\mathfrak{F},K}^{\operatorname{np}}$ and $\mathbf{A}_{\mathfrak{F},K}^{\operatorname{np},\operatorname{c}}$ for the \'etale $(\varphi,\Gamma_{\mathfrak{F},K})$-modules of $p$-adic false Tate curve extension $K\left(\pi_K^{1/p^\infty},\zeta_{p^\infty}\right)$. Various properties of these rings are studied and as applications, we show that $(\varphi,\Gamma_{\mathfrak{F},K})$-modules over these rings bridge $(\varphi,\Gamma)$-modules and $(\varphi,\tau)$-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 $(\varphi,\Gamma_{\mathfrak{F},K})$-modules over these rings.