Lemma 32.7.1. Let $f : X \to S$ be a morphism of quasi-compact and quasi-separated schemes. Then there exists a direct set $I$ and an inverse system $(f_ i : X_ i \to S_ i)$ of morphisms schemes over $I$, such that the transition morphisms $X_ i \to X_{i'}$ and $S_ i \to S_{i'}$ are affine, such that $X_ i$ and $S_ i$ are of finite type over $\mathbf{Z}$, and such that $(X \to S) = \mathop{\mathrm{lim}}\nolimits (X_ i \to S_ i)$.
32.7 Relative approximation
We discuss variants of Proposition 32.5.4 over a base.
Proof. Write $X = \mathop{\mathrm{lim}}\nolimits _{a \in A} X_ a$ and $S = \mathop{\mathrm{lim}}\nolimits _{b \in B} S_ b$ as in Proposition 32.5.4, i.e., with $X_ a$ and $S_ b$ of finite type over $\mathbf{Z}$ and with affine transition morphisms.
Fix $b \in B$. By Proposition 32.6.1 applied to $S_ b$ and $X = \mathop{\mathrm{lim}}\nolimits X_ a$ over $\mathbf{Z}$ we find there exists an $a \in A$ and a morphism $f_{a, b} : X_ a \to S_ b$ making the diagram
\[ \xymatrix{ X \ar[d] \ar[r] & S \ar[d] \\ X_ a \ar[r] & S_ b } \]
commute. Let $I$ be the set of triples $(a, b, f_{a, b})$ we obtain in this manner.
Let $(a, b, f_{a, b})$ and $(a', b', f_{a', b'})$ be in $I$. Let $b'' \leq \min (b, b')$. By Proposition 32.6.1 again, there exists an $a'' \geq \max (a, a')$ such that the compositions $X_{a''} \to X_ a \to S_ b \to S_{b''}$ and $X_{a''} \to X_{a'} \to S_{b'} \to S_{b''}$ are equal. We endow $I$ with the preorder
\[ (a, b, f_{a, b}) \geq (a', b', f_{a', b'}) \Leftrightarrow a \geq a',\ b \geq b',\text{ and } g_{b, b'} \circ f_{a, b} = f_{a', b'} \circ h_{a, a'} \]
where $h_{a, a'} : X_ a \to X_{a'}$ and $g_{b, b'} : S_ b \to S_{b'}$ are the transition morphisms. The remarks above show that $I$ is directed and that the maps $I \to A$, $(a, b, f_{a, b}) \mapsto a$ and $I \to B$, $(a, b, f_{a, b})$ are cofinal. If for $i = (a, b, f_{a, b})$ we set $X_ i = X_ a$, $S_ i = S_ b$, and $f_ i = f_{a, b}$, then we get an inverse system of morphisms over $I$ and we have
\[ \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i = \mathop{\mathrm{lim}}\nolimits _{a \in A} X_ a = X \quad \text{and}\quad \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i = \mathop{\mathrm{lim}}\nolimits _{b \in B} S_ b = S \]
by Categories, Lemma 4.17.4 (recall that limits over $I$ are really limits over the opposite category associated to $I$ and hence cofinal turns into initial). This finishes the proof. $\square$
Lemma 32.7.2. Let $f : X \to S$ be a morphism of schemes. Assume that
$X$ is quasi-compact and quasi-separated, and
$S$ is quasi-separated.
Then $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a limit of a directed system of schemes $X_ i$ of finite presentation over $S$ with affine transition morphisms over $S$.
Proof. Since $f(X)$ is quasi-compact we may replace $S$ by a quasi-compact open containing $f(X)$. Hence we may assume $S$ is quasi-compact. By Lemma 32.7.1 we can write $(X \to S) = \mathop{\mathrm{lim}}\nolimits (X_ i \to S_ i)$ for some directed inverse system of morphisms of finite type schemes over $\mathbf{Z}$ with affine transition morphisms. Since limits commute with limits (Categories, Lemma 4.14.10) we have $X = \mathop{\mathrm{lim}}\nolimits X_ i \times _{S_ i} S$. Let $i \geq i'$ in $I$. The morphism $X_ i \times _{S_ i} S \to X_{i'} \times _{S_{i'}} S$ is affine as the composition
\[ X_ i \times _{S_ i} S \to X_ i \times _{S_{i'}} S \to X_{i'} \times _{S_{i'}} S \]
where the first morphism is a closed immersion (by Schemes, Lemma 26.21.9) and the second is a base change of an affine morphism (Morphisms, Lemma 29.11.8) and the composition of affine morphisms is affine (Morphisms, Lemma 29.11.7). The morphisms $f_ i$ are of finite presentation (Morphisms, Lemmas 29.21.9 and 29.21.11) and hence the base changes $X_ i \times _{f_ i, S_ i} S \to S$ are of finite presentation (Morphisms, Lemma 29.21.4). $\square$
Lemma 32.7.3. Let $X \to S$ be an integral morphism with $S$ quasi-compact and quasi-separated. Then $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i \to S$ finite and of finite presentation.
Proof. Consider the sheaf $\mathcal{A} = f_*\mathcal{O}_ X$. This is a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras, see Schemes, Lemma 26.24.1. Combining Properties, Lemma 28.22.13 we can write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{A}_ i$ as a filtered colimit of finite and finitely presented $\mathcal{O}_ S$-algebras. Then
\[ X_ i = \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}_ i) \longrightarrow S \]
is a finite and finitely presented morphism of schemes. By construction $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$ which proves the lemma. $\square$
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