The Stacks project

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 $X_{a''} \to X_ a \to S_ b$ and $X_{a''} \to X_{a'} \to S_{b'} \to S_ b$ 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 29.21.9 and 29.21.11) and hence the base changes $X_ a \times _{f_{a, b}, S_ b} S \to S$ are of finite presentation (Morphisms, Lemma 29.21.4). $\square$

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