The Stacks project

87.13 Colimits of algebraic spaces along thickenings

A special type of formal algebraic space is one which can globally be written as a cofiltered colimit of algebraic spaces along thickenings as in the following lemma. We will see later (in Section 87.18) that any quasi-compact and quasi-separated formal algebraic space is such a global colimit.

Lemma 87.13.1. Let $S$ be a scheme. Suppose given a directed set $\Lambda $ and a system of algebraic spaces $(X_\lambda , f_{\lambda \mu })$ over $\Lambda $ where each $f_{\lambda \mu } : X_\lambda \to X_\mu $ is a thickening. Then $X = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } X_\lambda $ is a formal algebraic space over $S$.

Proof. Since we take the colimit in the category of fppf sheaves, we see that $X$ is a sheaf. Choose and fix $\lambda \in \Lambda $. Choose an étale covering $\{ X_{i, \lambda } \to X_\lambda \} $ where $X_ i$ is an affine scheme over $S$, see Properties of Spaces, Lemma 66.6.1. For each $\mu \geq \lambda $ there exists a cartesian diagram

\[ \xymatrix{ X_{i, \lambda } \ar[r] \ar[d] & X_{i, \mu } \ar[d] \\ X_\lambda \ar[r] & X_\mu } \]

with étale vertical arrows, see More on Morphisms of Spaces, Theorem 76.8.1 (this also uses that a thickening is a surjective closed immersion which satisfies the conditions of the theorem). Moreover, these diagrams are unique up to unique isomorphism and hence $X_{i, \mu } = X_\mu \times _{X_{\mu '}} X_{i, \mu '}$ for $\mu ' \geq \mu $. The morphisms $X_{i, \mu } \to X_{i, \mu '}$ is a thickening as a base change of a thickening. Each $X_{i, \mu }$ is an affine scheme by Limits of Spaces, Proposition 70.15.2 and the fact that $X_{i, \lambda }$ is affine. Set $X_ i = \mathop{\mathrm{colim}}\nolimits _{\mu \geq \lambda } X_{i, \mu }$. Then $X_ i$ is an affine formal algebraic space. The morphism $X_ i \to X$ is étale because given an affine scheme $U$ any $U \to X$ factors through $X_\mu $ for some $\mu \geq \lambda $ (details omitted). In this way we see that $X$ is a formal algebraic space. $\square$

Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. How does one prove or check that $X$ is a global colimit as in Lemma 87.13.1? To do this we look for maps $i : Z \to X$ where $Z$ is an algebraic space over $S$ and $i$ is surjective and a closed immersion, in other words, $i$ is a thickening. This makes sense as $i$ is representable by algebraic spaces (Lemma 87.11.3) and we can use Bootstrap, Definition 80.4.1 as before.

Example 87.13.2. Let $(A, \mathfrak m, \kappa )$ be a valuation ring, which is $(\pi )$-adically complete for some nonzero $\pi \in \mathfrak m$. Assume also that $\mathfrak m$ is not finitely generated. An example is $A = \mathcal{O}_{\mathbf{C}_ p}$ and $\pi = p$ where $\mathcal{O}_{\mathbf{C}_ p}$ is the ring of integers of the field of $p$-adic complex numbers $\mathbf{C}_ p$ (this is the completion of the algebraic closure of $\mathbf{Q}_ p$). Another example is

\[ A = \left\{ \sum \nolimits _{\alpha \in \mathbf{Q},\ \alpha \geq 0} a_\alpha t^\alpha \middle | \begin{matrix} a_\alpha \in \kappa \text{ and for all }n\text{ there are only a} \\ \text{finite number of nonzero }a_\alpha \text{ with }\alpha \leq n \end{matrix} \right\} \]

and $\pi = t$. Then $X = \text{Spf}(A)$ is an affine formal algebraic space and $\mathop{\mathrm{Spec}}(\kappa ) \to X$ is a thickening which corresponds to the weak ideal of definition $\mathfrak m \subset A$ which is however not an ideal of definition.

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 $\mathfrak X$.

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

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

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 $\mathfrak X$.

Comments (2)

Comment #1946 by Brian Conrad on

In the final Remark you write "If is an admissible topological ring then ", so remove the second "". The statement about not getting the same correspondence as in the preceding Remark will be easier to understand (in terms of what can go wrong, with the subtlety of "ideal of definition" vs. McQuillan's more robust notion involving just openness and topological nilpotence of elements) if an Example is given to illustrate the issue. Good luck making an example.

Comment #2000 by on

Thanks, for the typo. Also, as for your question, I have added an example provided by David Hansen here.

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