The Stacks project

Remark 86.9.8. The classical affine formal algebraic spaces correspond to the affine formal schemes considered in EGA ([EGA]). To explain this we assume our base scheme is $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Let $\mathfrak X = \text{Spf}(A)$ be an affine formal scheme. Let $h_\mathfrak X$ be its functor of points as in Lemma 86.2.1. Then $h_\mathfrak X = \mathop{\mathrm{colim}}\nolimits h_{\mathop{\mathrm{Spec}}(A/I)}$ where the colimit is over the collection of ideals of definition of the admissible topological ring $A$. This follows from ( when evaluating on affine schemes and it suffices to check on affine schemes as both sides are fppf sheaves, see Lemma 86.2.2. Thus $h_\mathfrak X$ is an affine formal algebraic space. In fact, it is a classical affine formal algebraic space by Definition 86.9.7. Thus Lemma 86.2.1 tells us the category of affine formal schemes is equivalent to the category of classical affine formal algebraic spaces.

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