The Stacks project

Remark 87.13.3 (Weak ideals of definition). Let $\mathfrak X$ be a formal scheme in the sense of McQuillan, see Remark 87.2.3. An weak ideal of definition for $\mathfrak X$ is an ideal sheaf $\mathcal{I} \subset \mathcal{O}_\mathfrak X$ such that for all $\mathfrak U \subset \mathfrak X$ affine formal open subscheme the ideal $\mathcal{I}(\mathfrak U) \subset \mathcal{O}_\mathfrak X(\mathfrak U)$ is a weak ideal of definition of the weakly admissible topological ring $\mathcal{O}_\mathfrak X(\mathfrak U)$. It suffices to check the condition on the members of an affine open covering. There is a one-to-one correspondence

This correspondence associates to $\mathcal{I}$ the scheme $Z = (\mathfrak X, \mathcal{O}_\mathfrak X/\mathcal{I})$ together with the obvious morphism to $\mathfrak X$. A fundamental system of weak ideals of definition is a collection of weak ideals of definition $\mathcal{I}_\lambda $ such that on every affine open formal subscheme $\mathfrak U \subset \mathfrak X$ the ideals

form a fundamental system of weak ideals of definition of the weakly admissible topological ring $A$. It suffices to check on the members of an affine open covering. We conclude that the formal algebraic space $h_\mathfrak X$ associated to the McQuillan formal scheme $\mathfrak X$ is a colimit of schemes as in Lemma 87.13.1 if and only if there exists a fundamental system of weak ideals of definition for $\mathfrak X$.