The Stacks project

Remark 81.13.2. In this remark we translate the statement and proof of Lemma 81.13.1 into the language of formal schemes à la EGA. Looking at Remark 81.8.4 we see that the lemma can be translated as follows

  • Every quasi-compact and quasi-separated formal scheme has a fundamental system of ideals of definition.

To prove this we first use the induction principle (reformulated for quasi-compact and quasi-separated formal schemes) of Cohomology of Schemes, Lemma 29.4.1 to reduce to the following situation: $\mathfrak X = \mathfrak U \cup \mathfrak V$ with $\mathfrak U$, $\mathfrak V$ open formal subschemes, with $\mathfrak V$ affine, and the result is true for $\mathfrak U$, $\mathfrak V$, and $\mathfrak U \cap \mathfrak V$. Pick any ideals of definition $\mathcal{I} \subset \mathcal{O}_\mathfrak U$ and $\mathcal{J} \subset \mathcal{O}_\mathfrak V$. By our assumption that we have a fundamental system of ideals of definition on $\mathfrak U$ and $\mathfrak V$ and because $\mathfrak U \cap \mathfrak V$ is quasi-compact, we can find ideals of definition $\mathcal{I}' \subset \mathcal{I}$ and $\mathcal{J}' \subset \mathcal{J}$ such that

\[ \mathcal{I}'|_{\mathfrak U \cap \mathfrak V} \subset \mathcal{J}|_{\mathfrak U \cap \mathfrak V} \quad \text{and}\quad \mathcal{J}'|_{\mathfrak U \cap \mathfrak V} \subset \mathcal{I}|_{\mathfrak U \cap \mathfrak V} \]

Let $U \to U' \to \mathfrak U$ and $V \to V' \to \mathfrak V$ be the closed immersions determined by the ideals of definition $\mathcal{I}' \subset \mathcal{I} \subset \mathcal{O}_\mathfrak U$ and $\mathcal{J}' \subset \mathcal{J} \subset \mathcal{O}_\mathfrak V$. Let $\mathfrak U \cap V$ denote the open subscheme of $V$ whose underlying topological space is that of $\mathfrak U \cap \mathfrak V$. By our choice of $\mathcal{I}'$ there is a factorization $\mathfrak U \cap V \to U'$. We define similarly $U \cap \mathfrak V$ which factors through $V'$. Then we consider

\[ Z_ U = \text{scheme theoretic image of } U \amalg (\mathfrak U \cap V) \longrightarrow U' \]

and

\[ Z_ V = \text{scheme theoretic image of } (U \cap \mathfrak V) \amalg V \longrightarrow V' \]

Since taking scheme theoretic images of quasi-compact morphisms commutes with restriction to opens (Morphisms, Lemma 28.6.3) we see that $Z_ U \cap \mathfrak V = \mathfrak U \cap Z_ V$. Thus $Z_ U$ and $Z_ V$ glue to a scheme $Z$ which comes equipped with a morphism $Z \to \mathfrak X$. Analogous to the discussion in Remark 81.8.3 we see that $Z$ corresponds to a weak ideal of definition $\mathcal{I}_ Z \subset \mathcal{O}_\mathfrak X$. Note that $Z_ U \subset U'$ and that $Z_ V \subset V'$. Thus the collection of all $\mathcal{I}_ Z$ constructed in this manner forms a fundamental system of weak ideals of definition. Hence a subfamily gives a fundamental system of ideals of definition, see Remark 81.8.4.


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