Lemma 37.69.2 (0CSJ): Derived Chow's lemma

Lemma 37.69.2 (Derived Chow's lemma). Let $A$ be a ring. Let $X$ be a separated scheme of finite presentation over $A$ . Let $x \in X$ . Then there exist an open neighbourhood $U \subset X$ of $x$ , an $n \geq 0$ , an open $V \subset \mathbf{P}^ n_ A$ , a closed subscheme $Z \subset X \times _ A \mathbf{P}^ n_ A$ , a point $z \in Z$ , 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,
$b : Z \to X$ is an isomorphism over $U$ and $b(z) = x$ ,
$c : Z \to \mathbf{P}^ n_ A$ is a closed immersion over $V$ ,
$b^{-1}(U) = c^{-1}(V)$ , in particular $c(z) \in V$ ,
$E|_{X \times _ A V} \cong (b, c)_*\mathcal{O}_ Z|_{X \times _ A V}$ ,
$E$ is pseudo-coherent and supported on $Z$ .

Proof. We can find a finite type $\mathbf{Z}$ -subalgebra $A' \subset A$ and a scheme $X'$ separated and of finite presentation over $A'$ whose base change to $A$ is $X$ . See Limits, Lemmas 32.10.1 and 32.8.6. Let $x' \in X'$ be the image of $x$ . If we can prove the lemma for $x' \in X'/A'$ , then the lemma follows for $x \in X/A$ . Namely, if $U', n', V', Z', z', E'$ provide the solution for $x' \in X'/A'$ , then we can let $U \subset X$ be the inverse image of $U'$ , let $n = n'$ , let $V \subset \mathbf{P}^ n_ A$ be the inverse image of $V'$ , let $Z \subset X \times \mathbf{P}^ n$ be the scheme theoretic inverse image of $Z'$ , let $z \in Z$ be the unique point mapping to $x$ , and let $E$ be the derived pullback of $E'$ . Observe that $E$ is pseudo-coherent by Cohomology, Lemma 20.47.3. It only remains to check (5). To see this set $W = b^{-1}(U) = c^{-1}(V)$ and $W' = (b')^{-1}(U) = (c')^{-1}(V')$ 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 37.69.1 the schemes $X \times _ A V$ and $W'$ are Tor independent over $X' \times _{A'} V'$ . Hence the derived pullback of $(b', c')_*\mathcal{O}_{W'}$ to $X \times _ A V$ is $(b, c)_*\mathcal{O}_ W$ by Derived Categories of Schemes, Lemma 36.22.5. 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 affine open neighbourhood $U \subset X$ of $x$ . 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

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

Since the projection $X \times \mathbf{P}^ n \to X$ is separated, we conclude from Morphisms, Lemma 29.6.8 that $b : Z \to X$ is an isomorphism over $U$ . Let $z \in Z$ be the unique point lying over $x$ .

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 the same lemma we used above. Choose $V \subset \mathbf{P}^ n_ A$ open with $V \cap Y = j(U)$ . Then we see that (3) and (4) hold.

Because $A$ is Noetherian we see that $X$ and $X \times _ A \mathbf{P}^ n_ A$ are Noetherian schemes. Hence we can take $E = (b, c)_*\mathcal{O}_ Z$ in this case, see Derived Categories of Schemes, Lemma 36.10.3. This finishes the proof. $\square$

The Stacks project

Comments (0)

Post a comment