## 32.7 Relative approximation

The title of this section refers to results of the following type.

Lemma 32.7.1. Let $f : X \to S$ be a morphism of schemes. Assume that

1. $X$ is quasi-compact and quasi-separated, and

2. $S$ 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 as well. Write $X = \mathop{\mathrm{lim}}\nolimits X_ a$ and $S = \mathop{\mathrm{lim}}\nolimits S_ b$ 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. By Proposition 32.6.1 we find that for each $b$ there exists an $a$ 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. Moreover the same proposition implies that, given a second triple $(a', b', f_{a', b'})$, there exists an $a'' \geq a'$ such that the compositions $X_{a''} \to X_ a \to X_ b$ and $X_{a''} \to X_{a'} \to X_{b'} \to X_ b$ are equal. Consider the set of triples $(a, b, f_{a, b})$ endowed with the preorder

$(a, b, f_{a, b}) \geq (a', b', f_{a', b'}) \Leftrightarrow a \geq a',\ b' \geq b,\text{ and } f_{a', b'} \circ h_{a, a'} = g_{b', b} \circ f_{a, b}$

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 this system is directed. It follows formally from the equalities $X = \mathop{\mathrm{lim}}\nolimits X_ a$ and $S = \mathop{\mathrm{lim}}\nolimits S_ b$ that

$X = \mathop{\mathrm{lim}}\nolimits _{(a, b, f_{a, b})} X_ a \times _{f_{a, b}, S_ b} S.$

where the limit is over our directed system above. The transition morphisms $X_ a \times _{S_ b} S \to X_{a'} \times _{S_{b'}} S$ are affine as the composition

$X_ a \times _{S_ b} S \to X_ a \times _{S_{b'}} S \to X_{a'} \times _{S_{b'}} S$

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 $f_{a, b}$ are of finite presentation (Morphisms, Lemmas 29.20.9 and 29.20.11) and hence the base changes $X_ a \times _{f_{a, b}, S_ b} S \to S$ are of finite presentation (Morphisms, Lemma 29.20.4). $\square$

Lemma 32.7.2. 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. $\square$

Comment #4915 by Arnab Kundu on

" Moreover the same proposition implies that, given a second triple (a′,b′,fa′,b′), there exists an a″≥a′ such that the compositions Xa″→Xa→Xb and Xa″→Xa′→Xb′→Xb are equal. " Should be replaced with "Moreover the same proposition implies that, given a second triple (a′,b′,fa′,b′), there exists an a″≥a′ such that the compositions Xa″→Xa→Sb and Xa″→Xa′→Sb′→Sb are equal."

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