The Stacks project

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

  1. $Z \to X \times _ A \mathbf{P}^ n_ A$ is of finite presentation,

  2. $b : Z \to X$ is an isomorphism over $U$ and $b(z) = x$,

  3. $c : Z \to \mathbf{P}^ n_ A$ is a closed immersion over $V$,

  4. $b^{-1}(U) = c^{-1}(V)$, in particular $c(z) \in V$,

  5. $E|_{X \times _ A V} \cong (b, c)_*\mathcal{O}_ Z|_{X \times _ A V}$,

  6. $E$ 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 simiarly 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. $\square$

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