## 32.7 Relative approximation

We discuss variants of Proposition 32.5.4 over a base.

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 $(X \to S) = \mathop{\mathrm{lim}}\nolimits (X_ i \to S_ i)$.

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

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. 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. $\square$

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."

Comment #6919 by Gerard Freixas on

I could not see how the first paragraph of the proof of Lemma 09MV is deduced from Proposition 01ZC. One can nevertheless apply Corollaire 8.13.2 in EGAIV, which I could not find in Chapter 32. Maybe it is worth including it?

Comment #6922 by on

Could you please try and say what is the first sentence of the proof that you had trouble with? I believe the proof is correct as written. Feel free to latex a version of the analogue of EGA IV 8.13.2 for the Stacks project and submit it to stacks.project@gmail.com

Comment #6928 by Gerard Freixas on

I see, I agree. You apply Prop 01ZC to $S_{b}\to\Spec Z$ and $T_i$ given by the system $X_{a}$. Sorry for the blindness!

Comment #6929 by on

Just to clarify what Gerard said in #6928: Yes, you apply 32.6.1 in the case where the base scheme is the spectrum of the integers to see that for all $b$ there is an $a$ and a morphism $f_{a, b} : X_a \to S_b$ compatible with $f$. Then you use 32.6.1 again, to see that in fact $f : X \to S$ is the limit of the morphisms $f_{a, b}$ for a suitable ordering on the triples $(a, b, f_{a, b})$. For clarity, I have added a lemma stating this intermediate result. I think splitting up the proof in this manner will also help people parse better what is going on. See this commit. (let me know if your name is spelled ok or if you want your name to be different in some way)

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