Lemma 76.51.2 (0CTR): Derived Chow's lemma

Lemma 76.51.2 (Derived Chow's lemma). Let $A$ be a ring. Let $X$ be a separated algebraic space of finite presentation over $A$ . Let $x \in |X|$ . Then there exist an $n \geq 0$ , a closed subspace $Z \subset X \times _ A \mathbf{P}^ n_ A$ , a point $z \in |Z|$ , an open $V \subset \mathbf{P}^ n_ A$ , and an object $E$ in $D(\mathcal{O}_{X \times _ A \mathbf{P}^ n_ A})$ such that

$Z \to X \times _ A \mathbf{P}^ n_ A$ is of finite presentation,
$c : Z \to \mathbf{P}^ n_ A$ is a closed immersion over $V$ , set $W = c^{-1}(V)$ ,
the restriction of $b : Z \to X$ to $W$ is étale, $z \in W$ , and $b(z) = x$ ,
$E|_{X \times _ A V} \cong (b, c)_*\mathcal{O}_ Z|_{X \times _ A V}$ ,
$E$ is pseudo-coherent and supported on $Z$ .

Proof. We can find a finite type $\mathbf{Z}$ -subalgebra $A' \subset A$ and an algebraic space $X'$ separated and of finite presentation over $A'$ whose base change to $A$ is $X$ . See Limits of Spaces, Lemmas 70.7.1 and 70.6.9. Let $x' \in |X'|$ be the image of $x$ . If we can prove the lemma for $(X'/A', x')$ , then the lemma follows for $(X/A, x)$ . Namely, if $n', Z', z', V', E'$ provide the solution for $(X'/A', x')$ , then we can let $n = n'$ , let $Z \subset X \times \mathbf{P}^ n$ be the inverse image of $Z'$ , let $z \in Z$ be the unique point mapping to $x$ , let $V \subset \mathbf{P}^ n_ A$ be the inverse image of $V'$ , and let $E$ be the derived pullback of $E'$ . Observe that $E$ is pseudo-coherent by Cohomology on Sites, Lemma 21.45.3. It only remains to check (5). To see this set $W = c^{-1}(V)$ and $W' = (c')^{-1}(V')$ and consider the cartesian square

$\xymatrix{ W \ar[d]_{(b, c)} \ar[r] & W' \ar[d]^{(b', c')} \\ X \times _ A V \ar[r] & X' \times _{A'} V' }$

By Lemma 76.51.1 $X \times _ A V$ and $W'$ are tor-independent over $X' \times _{A'} V'$ . Thus the derived pullback of $(b', c')_*\mathcal{O}_{W'}$ to $X \times _ A V$ is $(b, c)_*\mathcal{O}_ W$ by Derived Categories of Spaces, Lemma 75.20.4. This also uses that $R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}$ because $(b', c')$ is a closed immersion and similarly for $(b, c)_*\mathcal{O}_ Z$ . Since $E'|_{U' \times _{A'} V'} = (b', c')_*\mathcal{O}_{W'}$ we obtain $E|_{U \times _ A V} = (b, c)_*\mathcal{O}_ W$ and (5) holds. This reduces us to the situation described in the next paragraph.

Assume $A$ is of finite type over $\mathbf{Z}$ . Choose an étale morphism $U \to X$ where $U$ is an affine scheme and a point $u \in U$ mapping to $x$ . Then $U$ is of finite type over $A$ . Choose a closed immersion $U \to \mathbf{A}^ n_ A$ and denote $j : U \to \mathbf{P}^ n_ A$ the immersion we get by composing with the open immersion $\mathbf{A}^ n_ A \to \mathbf{P}^ n_ A$ . Let $Z$ be the scheme theoretic closure of

$(\text{id}_ U, j) : U \longrightarrow X \times _ A \mathbf{P}^ n_ A$

Let $z \in Z$ be the image of $u$ . Let $Y \subset \mathbf{P}^ n_ A$ be the scheme theoretic closure of $j$ . Then it is clear that $Z \subset X \times _ A Y$ is the scheme theoretic closure of $(\text{id}_ U, j) : U \to X \times _ A Y$ . As $X$ is separated, the morphism $X \times _ A Y \to Y$ is separated as well. Hence we see that $Z \to Y$ is an isomorphism over the open subscheme $j(U) \subset Y$ by Morphisms of Spaces, Lemma 67.16.7. Choose $V \subset \mathbf{P}^ n_ A$ open with $V \cap Y = j(U)$ . Then we see that (2) holds, that $W = (\text{id}_ U, j)(U)$ , and hence that (3) holds. Part (1) holds because $A$ is Noetherian.

Because $A$ is Noetherian we see that $X$ and $X \times _ A \mathbf{P}^ n_ A$ are Noetherian algebraic spaces. Hence we can take $E = (b, c)_*\mathcal{O}_ Z$ in this case: (4) is clear and for (5) see Derived Categories of Spaces, Lemma 75.13.7. This finishes the proof. $\square$

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