Fix a field $k$ with a complete and non-archimedean absolute value $|\raisebox{1.72218pt}{.}|$. The valuation ring of $k$ and the maximal ideal of the valuation ring are denoted by $\mathfrak{o}_{k}$ and $\mathfrak{m}_{k}$, respectively, that is,

$$\mathfrak{o}_{k}:=\{a\in k\mid|a|\leq 1\}\quad\text{and}\quad\mathfrak{m}_{k}:=\{x\in k\mid|x|<1\}.$$

In the case where $|\raisebox{1.72218pt}{.}|$ is discrete, we fix a uniformizing parameter $\varpi$ of $\mathfrak{m}_{k}$, that is, $\mathfrak{m}_{k}=\varpi\mathfrak{o}_{k}$.