The Stacks project

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. $\square$


Comments (1)

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."


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09MU. Beware of the difference between the letter 'O' and the digit '0'.