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

$\{ \text{weak ideals of definition for }\mathfrak X\} \leftrightarrow \{ \text{thickenings }i : Z \to h_\mathfrak X\text{ as above}\}$

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

$I_\lambda = \mathcal{I}_\lambda (\mathfrak U) \subset A = \Gamma (\mathfrak U, \mathcal{O}_\mathfrak X)$

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 $\mathfrak X$.

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