(1.6) If $\mathscr{X}$ is a Noetherian $R$-scheme and $C$ is a subscheme of $\mathscr{X}_{k}$, then we will denote by $\widehat{\mathscr{X}_{/C}}$ the formal completion of $\mathscr{X}$ along $C$. If $\mathscr{X}$ is of finite type over $R$, then $\widehat{\mathscr{X}_{/C}}$ is formally of finite type over $R$ (or special, in the terminology of [Be96]). That is, it has a finite cover by open formal subschemes of the form $\mathrm{Spf}\,(A)$ where $A$ is a quotient of a topological $R$-algebra of the form $R\{x_{1},\ldots,x_{m}\}[\negthinspace[y_{1},\ldots,y_{n}]\negthinspace]$. Every Noetherian formal scheme $\mathfrak{X}$ has a unique maximal ideal of definition $\mathscr{I}$, consisting of all the topologically nilpotent elements in $\mathcal{O}_{\mathfrak{X}}$. The closed subscheme of $\mathfrak{X}$ defined by $\mathscr{I}$ will be denoted by $\mathfrak{X}_{\mathrm{red}}$. This construction induces a functor from the category of Noetherian formal schemes to the category of reduced Noetherian schemes. If $\mathfrak{X}$ is a scheme, then $\mathfrak{X}_{\mathrm{red}}$ is the maximal reduced closed subscheme of $\mathfrak{X}$.