Lemma 70.10.1. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated algebraic spaces over $\mathbf{Z}$. Then there exists a direct set $I$ and an inverse system $(f_ i : X_ i \to Y_ i)$ of morphisms algebraic spaces over $I$, such that the transition morphisms $X_ i \to X_{i'}$ and $Y_ i \to Y_{i'}$ are affine, such that $X_ i$ and $Y_ i$ are quasi-separated and of finite type over $\mathbf{Z}$, and such that $(X \to Y) = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$.
70.10 Relative approximation
We discuss variants of Proposition 70.8.1 over a base.
Proof. Write $X = \mathop{\mathrm{lim}}\nolimits _{a \in A} X_ a$ and $Y = \mathop{\mathrm{lim}}\nolimits _{b \in B} Y_ b$ as in Proposition 70.8.1, i.e., with $X_ a$ and $Y_ b$ quasi-separated and of finite type over $\mathbf{Z}$ and with affine transition morphisms.
Fix $b \in B$. By Lemma 70.4.5 applied to $Y_ b$ and $X = \mathop{\mathrm{lim}}\nolimits X_ a$ over $\mathbf{Z}$ we find there exists an $a \in A$ and a morphism $f_{a, b} : X_ a \to Y_ b$ making the diagram
\[ \xymatrix{ X \ar[d] \ar[r] & Y \ar[d] \\ X_ a \ar[r] & Y_ b } \]
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 Lemma 70.4.5 again, there exists an $a'' \geq \max (a, a')$ such that the compositions $X_{a''} \to X_ a \to Y_ b \to Y_{b''}$ and $X_{a''} \to X_{a'} \to Y_{b'} \to Y_{b''}$ 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 $g_{b, b'} : Y_ b \to Y_{b'}$ 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 $X_ i = X_ a$, $Y_ i = Y_ b$, and $f_ i = f_{a, b}$, then we get an inverse system of morphisms over $I$ and we have
\[ \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i = \mathop{\mathrm{lim}}\nolimits _{a \in A} X_ a = X \quad \text{and}\quad \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i = \mathop{\mathrm{lim}}\nolimits _{b \in B} Y_ b = Y \]
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 70.10.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that
$X$ is quasi-compact and quasi-separated, and
$Y$ is quasi-separated.
Then $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a limit of a directed inverse system of algebraic spaces $X_ i$ of finite presentation over $Y$ with affine transition morphisms over $Y$.
Proof. Since $|f|(|X|)$ is quasi-compact we may replace $Y$ by a quasi-compact open subspace whose set of points contains $|f|(|X|)$. Hence we may assume $Y$ is quasi-compact as well. By Lemma 70.10.1 we can write $(X \to Y) = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$ 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 $X = \mathop{\mathrm{lim}}\nolimits X_ i \times _{Y_ i} Y$. For $i \geq i'$ the transition morphism $X_ i \times _{Y_ i} Y \to X_{i'} \times _{Y_{i'}} Y$ is affine as the composition
\[ X_ i \times _{Y_ i} Y \to X_ i \times _{Y_{i'}} Y \to X_{i'} \times _{Y_{i'}} Y \]
where the first morphism is a closed immersion (by Morphisms of Spaces, Lemma 67.4.5) and the second is a base change of an affine morphism (Morphisms of Spaces, Lemma 67.20.5) and the composition of affine morphisms is affine (Morphisms of Spaces, Lemma 67.20.4). The morphisms $f_ i$ are of finite presentation (Morphisms of Spaces, Lemmas 67.28.7 and 67.28.9) and hence the base changes $X_ i \times _{f_ i, Y_ i} Y \to Y$ are of finite presentation (Morphisms of Spaces, Lemma 67.28.3). $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)