The Stacks project

75.40 Chow's lemma

In this section we prove Chow's lemma (Lemma 75.40.5). We encourage the reader to take a look at Cohomology of Spaces, Section 68.18 for a weak version of Chow's lemma that is easy to prove and sufficient for many applications.

Since we have yet to define projective morphisms of algebraic spaces, the statements of lemmas (see for example Lemma 75.40.2) will involve representable proper morphisms, rather than projective ones.

Lemma 75.40.1. Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \to X_1$ and $U \to X_2$ be open immersions of algebraic spaces over $Y$ and assume $U$, $X_1$, $X_2$ of finite type and separated over $Y$. Then there exists a commutative diagram

\[ \xymatrix{ X_1' \ar[d] \ar[r] & X & X_2' \ar[l] \ar[d] \\ X_1 & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & X_2 } \]

of algebraic spaces over $Y$ where $X_ i' \to X_ i$ is a $U$-admissible blowup, $X_ i' \to X$ is an open immersion, and $X$ is separated and finite type over $Y$.

Proof. Throughout the proof all the algebraic spaces will be separated of finite type over $Y$. This in particular implies these algebraic spaces are quasi-compact and quasi-separated and that the morphisms between them will be quasi-compact and separated. See Morphisms of Spaces, Sections 66.4 and 66.8. We will use that if $U \to W$ is an immersion of such spaces over $Y$, then the scheme theoretic image $Z$ of $U$ in $W$ is a closed subspace of $W$ and $U \to Z$ is an open immersion, $U \subset Z$ is scheme theoretically dense, and $|U| \subset |Z|$ is dense. See Morphisms of Spaces, Lemma 66.17.7.

Let $X_{12} \subset X_1 \times _ Y X_2$ be the scheme theoretic image of $U \to X_1 \times _ Y X_2$. The projections $p_ i : X_{12} \to X_ i$ induce isomorphisms $p_ i^{-1}(U) \to U$ by Morphisms of Spaces, Lemma 66.16.7. Choose a $U$-admissible blowup $X_ i^ i \to X_ i$ such that the strict transform $X_{12}^ i$ of $X_{12}$ is isomorphic to an open subspace of $X_ i^ i$, see Lemma 75.39.4. Let $\mathcal{I}_ i \subset \mathcal{O}_{X_ i}$ be the corresponding finite type quasi-coherent sheaf of ideals. Recall that $X_{12}^ i \to X_{12}$ is the blowup in $p_ i^{-1}\mathcal{I}_ i \mathcal{O}_{X_{12}}$, see Divisors on Spaces, Lemma 70.18.3. Let $X_{12}'$ be the blowup of $X_{12}$ in $p_1^{-1}\mathcal{I}_1 p_2^{-1}\mathcal{I}_2 \mathcal{O}_{X_{12}}$, see Divisors on Spaces, Lemma 70.17.10 for what this entails. We obtain a commutative diagram

\[ \xymatrix{ X_{12}' \ar[d] \ar[r] & X_{12}^2 \ar[d] \\ X_{12}^1 \ar[r] & X_{12} } \]

where all the morphisms are $U$-admissible blowing ups. Since $X_{12}^ i \subset X_ i^ i$ is an open we may choose a $U$-admissible blowup $X_ i' \to X_ i^ i$ restricting to $X_{12}' \to X_{12}^ i$, see Divisors on Spaces, Lemma 70.19.3. Then $X_{12}' \subset X_ i'$ is an open subspace and the diagram

\[ \xymatrix{ X_{12}' \ar[d] \ar[r] & X_ i' \ar[d] \\ X_{12}^ i \ar[r] & X_ i^ i } \]

is commutative with vertical arrows blowing ups and horizontal arrows open immersions. Note that $X'_{12} \to X_1' \times _ Y X_2'$ is an immersion and proper (use that $X'_{12} \to X_{12}$ is proper and $X_{12} \to X_1 \times _ Y X_2$ is closed and $X_1' \times _ Y X_2' \to X_1 \times _ Y X_2$ is separated and apply Morphisms of Spaces, Lemma 66.40.6). Thus $X'_{12} \to X_1' \times _ Y X_2'$ is a closed immersion. If we define $X$ by glueing $X_1'$ and $X_2'$ along the common open subspace $X_{12}'$, then $X \to Y$ is of finite type and separated1. As compositions of $U$-admissible blowups are $U$-admissible blowups (Divisors on Spaces, Lemma 70.19.2) the lemma is proved. $\square$

Lemma 75.40.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $U \subset X$ be an open subspace. Assume

  1. $U$ is quasi-compact,

  2. $Y$ is quasi-compact and quasi-separated,

  3. there exists an immersion $U \to \mathbf{P}^ n_ Y$ over $Y$,

  4. $f$ is of finite type and separated.

Then there exists a commutative diagram

\[ \xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & Y & \mathbf{P}^ n_ Y \ar[l] } \]

where the arrows with source $U$ are open immersions, $X' \to X$ is a $U$-admissible blowup, $X' \to Z'$ is an open immersion, $Z' \to Y$ is a proper and representable morphism of algebraic spaces. More precisely, $Z' \to Z$ is a $U$-admissible blowup and $Z \to \mathbf{P}^ n_ Y$ is a closed immersion.

Proof. Let $Z \subset \mathbf{P}^ n_ Y$ be the scheme theoretic image of the immersion $U \to \mathbf{P}^ n_ Y$. Since $U \to \mathbf{P}^ n_ Y$ is quasi-compact we see that $U \subset Z$ is a (scheme theoretically) dense open subspace (Morphisms of Spaces, Lemma 66.17.7). Apply Lemma 75.40.1 to find a diagram

\[ \xymatrix{ X' \ar[d] \ar[r] & \overline{X}' & Z' \ar[l] \ar[d] \\ X & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & Z } \]

with properties as listed in the statement of that lemma. As $X' \to X$ and $Z' \to Z$ are $U$-admissible blowups we find that $U$ is a scheme theoretically dense open of both $X'$ and $Z'$ (see Divisors on Spaces, Lemmas 70.17.4 and 70.6.4). Since $Z' \to Z \to Y$ is proper we see that $Z' \subset \overline{X}'$ is a closed subspace (see Morphisms of Spaces, Lemma 66.40.6). It follows that $X' \subset Z'$ (scheme theoretically), hence $X'$ is an open subspace of $Z'$ (small detail omitted) and the lemma is proved. $\square$

Lemma 75.40.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated, of finite type, and $Y$ Noetherian. Then there exists a dense open subspace $U \subset X$ and a commutative diagram

\[ \xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & Y & \mathbf{P}^ n_ Y \ar[l] } \]

where the arrows with source $U$ are open immersions, $X' \to X$ is a $U$-admissible blowup, $X' \to Z'$ is an open immersion, $Z' \to Y$ is a proper and representable morphism of algebraic spaces. More precisely, $Z' \to Z$ is a $U$-admissible blowup and $Z \to \mathbf{P}^ n_ Y$ is a closed immersion.

Proof. By Limits of Spaces, Lemma 69.13.3 there exists a dense open subspace $U \subset X$ and an immersion $U \to \mathbf{A}^ n_ Y$ over $Y$. Composing with the open immersion $\mathbf{A}^ n_ Y \to \mathbf{P}^ n_ Y$ we obtain a situation as in Lemma 75.40.2 and the result follows. $\square$

Remark 75.40.4. In Lemmas 75.40.2 and 75.40.3 the morphism $g : Z' \to Y$ is a composition of projective morphisms. Presumably (by the analogue for algebraic spaces of Morphisms, Lemma 29.37.8) there exists a $g$-ample invertible sheaf on $Z'$. If we ever need this, then we will state and prove this here.

The following result is [IV Theorem 3.1, Kn]. Note that the immersion $X' \to \mathbf{P}^ n_ Y$ is quasi-compact, hence can be factored as $X' \to Z' \to \mathbf{P}^ n_ Y$ where the first morphism is an open immersion and the second morphism a closed immersion (Morphisms of Spaces, Lemma 66.17.7).

reference

Lemma 75.40.5 (Chow's lemma). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated of finite type, and $Y$ separated and Noetherian. Then there exists a commutative diagram

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

where $X' \to X$ is a $U$-admissible blowup for some dense open $U \subset X$ and the morphism $X' \to \mathbf{P}^ n_ Y$ is an immersion.

Proof. In this first paragraph of the proof we reduce the lemma to the case where $Y$ is of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. We may and do replace the base scheme $S$ by $\mathop{\mathrm{Spec}}(\mathbf{Z})$. We can write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ as a directed limit of separated algebraic spaces of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Limits of Spaces, Proposition 69.8.1 and Lemma 69.5.9. For all $i$ sufficiently large we can find a separated finite type morphism $X_ i \to Y_ i$ such that $X = Y \times _{Y_ i} X_ i$, see Limits of Spaces, Lemmas 69.7.1 and 69.6.9. Let $\eta _1, \ldots , \eta _ n$ be the generic points of the irreducible components of $|X|$ ($X$ is Noetherian as a finite type separated algebraic space over the Noetherian algebraic space $Y$ and therefore $|X|$ is a Noetherian topological space). By Limits of Spaces, Lemma 69.5.2 we find that the images of $\eta _1, \ldots , \eta _ n$ in $|X_ i|$ are distinct for $i$ large enough. We may replace $X_ i$ by the scheme theoretic image of the (quasi-compact, in fact affine) morphism $X \to X_ i$. After this replacement we see that the images of $\eta _1, \ldots , \eta _ n$ in $|X_ i|$ are the generic points of the irreducible components of $|X_ i|$, see Morphisms of Spaces, Lemma 66.16.3. Having said this, suppose we can find a diagram

\[ \xymatrix{ X_ i \ar[rd] & X_ i' \ar[l] \ar[d] \ar[r] & \mathbf{P}^ n_{Y_ i} \ar[ld] \\ & Y } \]

where $X_ i' \to X_ i$ is a $U_ i$-admissible blowup for some dense open $U_ i \subset X_ i$ and the morphism $X_ i' \to \mathbf{P}^ n_{Y_ i}$ is an immersion. Then the strict transform $X' \to X$ of $X$ relative to $X_ i' \to X_ i$ is a $U$-admissible blowing up where $U \subset X$ is the inverse image of $U_ i$ in $X$. Because of our carefuly chosen index $i$ it follows that $\eta _1, \ldots , \eta _ n \in |U|$ and $U \subset X$ is dense. Moreover, $X' \to \mathbf{P}^ n_ Y$ is an immersion as $X'$ is closed in $X_ i' \times _{X_ i} X = X_ i' \times _{Y_ i} Y$ which comes with an immersion into $\mathbf{P}^ n_ Y$. Thus we have reduced to the situation of the following paragraph.

Assume that $Y$ is separated of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Then $X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated of finite type as well. We apply Lemma 75.40.3 to $X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ to find a dense open subspace $U \subset X$ and a commutative diagram

\[ \xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & \mathop{\mathrm{Spec}}(\mathbf{Z}) & \mathbf{P}^ n_\mathbf {Z} \ar[l] } \]

with all the properties listed in the lemma. Note that $Z$ has an ample invertible sheaf, namely $\mathcal{O}_{\mathbf{P}^ n}(1)|_ Z$. Hence $Z' \to Z$ is a H-projective morphism by Morphisms, Lemma 29.43.16. It follows that $Z' \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is H-projective by Morphisms, Lemma 29.43.7. Thus there exists a closed immersion $Z' \to \mathbf{P}^ m_{\mathop{\mathrm{Spec}}(\mathbf{Z})}$ for some $m \geq 0$. It follows that the diagonal morphism

\[ X' \to Y \times \mathbf{P}^ m_\mathbf {Z} = \mathbf{P}^ m_ Y \]

is an immersion (because the composition with the projection to $\mathbf{P}^ m_\mathbf {Z}$ is an immersion) and we win. $\square$

[1] Because we may check closedness of the diagonal $X \to X \times _ Y X$ over the four open parts $X'_ i \times _ Y X'_ j$ of $X \times _ Y X$ where it is clear.

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