The Stacks project

Remark 87.13.4 (Ideals of definition). Let $\mathfrak X$ be a formal scheme à la EGA. An 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 an ideal of definition of the admissible topological ring $\mathcal{O}_\mathfrak X(\mathfrak U)$. It suffices to check the condition on the members of an affine open covering. We do not get the same correspondence between ideals of definition and thickenings $Z \to h_\mathfrak X$ as in Remark 87.13.3; an example is given in Example 87.13.2. A fundamental system of ideals of definition is a collection of 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 ideals of definition of the admissible topological ring $A$. It suffices to check on the members of an affine open covering. Suppose that $\mathfrak X$ is quasi-compact and that $\{ \mathcal{I}_\lambda \} _{\lambda \in \Lambda }$ is a fundamental system of weak ideals of definition. If $A$ is an admissible topological ring then all sufficiently small open ideals are ideals of definition (namely any open ideal contained in an ideal of definition is an ideal of definition). Thus since we only need to check on the finitely many members of an affine open covering we see that $\mathcal{I}_\lambda $ is an ideal of definition for $\lambda $ sufficiently large. Using the discussion in Remark 87.13.3 we conclude that the formal algebraic space $h_\mathfrak X$ associated to the quasi-compact formal scheme $\mathfrak X$ à la EGA is a colimit of schemes as in Lemma 87.13.1 if and only if there exists a fundamental system of ideals of definition for $\mathfrak X$.