The Stacks project

Remark 87.11.4. Modulo set theoretic issues the category of formal schemes à la EGA (see Section 87.2) is equivalent to a full subcategory of the category of formal algebraic spaces. To explain this we assume our base scheme is $\mathop{\mathrm{Spec}}(\mathbf{Z})$. By Lemma 87.2.2 the functor of points $h_\mathfrak X$ associated to a formal scheme $\mathfrak X$ is a sheaf in the fppf topology. By Lemma 87.2.1 the assignment $\mathfrak X \mapsto h_\mathfrak X$ is a fully faithful embedding of the category of formal schemes into the category of fppf sheaves. Given a formal scheme $\mathfrak X$ we choose an open covering $\mathfrak X = \bigcup \mathfrak X_ i$ with $\mathfrak X_ i$ affine formal schemes. Then $h_{\mathfrak X_ i}$ is an affine formal algebraic space by Remark 87.9.8. The morphisms $h_{\mathfrak X_ i} \to h_\mathfrak X$ are representable and open immersions. Thus $\{ h_{\mathfrak X_ i} \to h_\mathfrak X\} $ is a family as in Definition 87.11.1 and we see that $h_\mathfrak X$ is a formal algebraic space.