The Stacks project

70.12 Approximating proper morphisms

Lemma 70.12.1. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$ with $Y$ quasi-compact and quasi-separated. Then $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of algebraic spaces $X_ i$ proper and of finite presentation over $Y$ and with transition morphisms and morphisms $X \to X_ i$ closed immersions.

Proof. By Proposition 70.11.7 we can find a closed immersion $X \to X'$ with $X'$ separated and of finite presentation over $Y$. By Lemma 70.11.4 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i \to X'$ a closed immersion of finite presentation. We claim that for all $i$ large enough the morphism $X_ i \to Y$ is proper which finishes the proof.

To prove this we may assume that $Y$ is an affine scheme, see Morphisms of Spaces, Lemma 67.40.2. Next, we use the weak version of Chow's lemma, see Cohomology of Spaces, Lemma 69.18.1, to find a diagram

\[ \xymatrix{ X' \ar[rd] & X'' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^ n_ Y \ar[dl] \\ & Y & } \]

where $X'' \to \mathbf{P}^ n_ Y$ is an immersion, and $\pi : X'' \to X'$ is proper and surjective. Denote $X'_ i \subset X''$, resp. $\pi ^{-1}(X)$ the scheme theoretic inverse image of $X_ i \subset X'$, resp. $X \subset X'$. Then $\mathop{\mathrm{lim}}\nolimits X'_ i = \pi ^{-1}(X)$. Since $\pi ^{-1}(X) \to Y$ is proper (Morphisms of Spaces, Lemmas 67.40.4), we see that $\pi ^{-1}(X) \to \mathbf{P}^ n_ Y$ is a closed immersion (Morphisms of Spaces, Lemmas 67.40.6 and 67.12.3). Hence for $i$ large enough we find that $X'_ i \to \mathbf{P}^ n_ Y$ is a closed immersion by Lemma 70.5.16. Thus $X'_ i$ is proper over $Y$. For such $i$ the morphism $X_ i \to Y$ is proper by Morphisms of Spaces, Lemma 67.40.7. $\square$

Lemma 70.12.2. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $\mathbf{Z}$ with $Y$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_ i : X_ i \to Y_ i)$ of morphisms of algebraic spaces over $I$, such that the transition morphisms $X_ i \to X_{i'}$ and $Y_ i \to Y_{i'}$ are affine, such that $f_ i$ is proper and of finite presentation, such that $Y_ i$ is of finite presentation over $\mathbf{Z}$, and such that $(X \to Y) = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$.

Proof. By Lemma 70.12.1 we can write $X = \mathop{\mathrm{lim}}\nolimits _{k \in K} X_ k$ with $X_ k \to Y$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition 70.8.1) we can write $Y = \mathop{\mathrm{lim}}\nolimits _{j \in J} Y_ j$ with $Y_ j$ of finite presentation over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to Y_ j$ of finite presentation with $X_ k \cong Y \times _{Y_ j} X_{k, j}$ as algebraic spaces over $Y$, see Lemma 70.7.1. After increasing $j$ we may assume $X_{k, j} \to Y_ j$ is proper, see Lemma 70.6.13. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to Y_ j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $Y_{j'} \to Y_ j$ whose base change to $Y$ gives the morphism $X_{k'} \to X_ k$ (follows again from Lemma 70.7.1). These morphisms form the transition morphisms of the system. Some details omitted. $\square$

Recall the scheme theoretic support of a finite type quasi-coherent module, see Morphisms of Spaces, Definition 67.15.4.

Lemma 70.12.3. Assumptions and notation as in Situation 70.6.1. Let $\mathcal{F}_0$ be a quasi-coherent $\mathcal{O}_{X_0}$-module. Denote $\mathcal{F}$ and $\mathcal{F}_ i$ the pullbacks of $\mathcal{F}_0$ to $X$ and $X_ i$. Assume

  1. $f_0$ is locally of finite type,

  2. $\mathcal{F}_0$ is of finite type,

  3. the scheme theoretic support of $\mathcal{F}$ is proper over $Y$.

Then the scheme theoretic support of $\mathcal{F}_ i$ is proper over $Y_ i$ for some $i$.

Proof. We may replace $X_0$ by the scheme theoretic support of $\mathcal{F}_0$. By Morphisms of Spaces, Lemma 67.15.2 this guarantees that $X_ i$ is the support of $\mathcal{F}_ i$ and $X$ is the support of $\mathcal{F}$. Then, if $Z \subset X$ denotes the scheme theoretic support of $\mathcal{F}$, we see that $Z \to X$ is a universal homeomorphism. We conclude that $X \to Y$ is proper as this is true for $Z \to Y$ by assumption, see Morphisms, Lemma 29.41.9. By Lemma 70.6.13 we see that $X_ i \to Y$ is proper for some $i$. Then it follows that the scheme theoretic support $Z_ i$ of $\mathcal{F}_ i$ is proper over $Y$ by Morphisms of Spaces, Lemmas 67.40.5 and 67.40.4. $\square$

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