The Stacks project

Lemma 32.18.1. Let $I$ be a directed set. Let $(f_ i : X_ i \to S_ i)$ be an inverse system of morphisms of schemes over $I$. Assume

  1. all the morphisms $S_{i'} \to S_ i$ are affine,

  2. all the schemes $S_ i$ are quasi-compact and quasi-separated,

  3. the morphisms $f_ i$ are of finite type, and

  4. the morphisms $X_{i'} \to X_ i \times _{S_ i} S_{i'}$ are closed immersions.

Let $f : X = \mathop{\mathrm{lim}}\nolimits _ i X_ i \to S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$ be the limit. Let $d \geq 0$. If every fibre of $f$ has dimension $\leq d$, then for some $i$ every fibre of $f_ i$ has dimension $\leq d$.

Proof. For each $i$ let $U_ i = \{ x \in X_ i \mid \dim _ x((X_ i)_{f_ i(x)}) \leq d\} $. This is an open subset of $X_ i$, see Morphisms, Lemma 29.28.4. Set $Z_ i = X_ i \setminus U_ i$ (with reduced induced scheme structure). We have to show that $Z_ i = \emptyset $ for some $i$. If not, then $Z = \mathop{\mathrm{lim}}\nolimits Z_ i \not= \emptyset $, see Lemma 32.4.3. Say $z \in Z$ is a point. Note that $Z \subset X$ is a closed subscheme. Set $s = f(z)$. For each $i$ let $s_ i \in S_ i$ be the image of $s$. We remark that $Z_ s$ is the limit of the schemes $(Z_ i)_{s_ i}$ and $Z_ s$ is also the limit of the schemes $(Z_ i)_{s_ i}$ base changed to $\kappa (s)$. Moreover, all the morphisms

\[ Z_ s \longrightarrow (Z_{i'})_{s_{i'}} \times _{\mathop{\mathrm{Spec}}(\kappa (s_{i'}))} \mathop{\mathrm{Spec}}(\kappa (s)) \longrightarrow (Z_ i)_{s_ i} \times _{\mathop{\mathrm{Spec}}(\kappa (s_ i))} \mathop{\mathrm{Spec}}(\kappa (s)) \longrightarrow X_ s \]

are closed immersions by assumption (4). Hence $Z_ s$ is the scheme theoretic intersection of the closed subschemes $(Z_ i)_{s_ i} \times _{\mathop{\mathrm{Spec}}(\kappa (s_ i))} \mathop{\mathrm{Spec}}(\kappa (s))$ in $X_ s$. Since all the irreducible components of the schemes $(Z_ i)_{s_ i} \times _{\mathop{\mathrm{Spec}}(\kappa (s_ i))} \mathop{\mathrm{Spec}}(\kappa (s))$ have dimension $> d$ and contain $z$ we conclude that $Z_ s$ contains an irreducible component of dimension $> d$ passing through $z$ which contradicts the fact that $Z_ s \subset X_ s$ and $\dim (X_ s) \leq d$. $\square$

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