Lemma 57.16.4. Let $k$ be a field. Let $S$ be a finite type scheme over $k$ with $k$-rational point $s$. Let $Y \to S$ be a smooth proper morphism. Let $X = Y_ s \times S \to S$ be the constant family with fibre $Y_ s$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Assume the restriction

\[ L(Y_ s \times _ S Y_ s \to X \times _ S Y)^*K \cong \Delta _{Y_ s/k, *} \mathcal{O}_{Y_ s} \]

in $D(\mathcal{O}_{Y_ s \times Y_ s})$. Then there is an open neighbourhood $s \in U \subset S$ such that $Y|_ U$ is isomorphic to $Y_ s \times U$ over $U$.

**Proof.**
Denote $i : Y_ s \times Y_ s = X_ s \times Y_ s \to X \times _ S Y$ the natural closed immersion. (We will write $Y_ s$ and not $X_ s$ for the fibre of $X$ over $s$ from now on.) Let $z \in Y_ s \times Y_ s = (X \times _ S Y)_ s \subset X \times _ S Y$ be a closed point. As indicated we think of $z$ both as a closed point of $Y_ s \times Y_ s$ as well as a closed point of $X \times _ S Y$.

Case I: $z \not\in \Delta _{Y_ s/k}(Y_ s)$. Denote $\mathcal{O}_ z$ the coherent $\mathcal{O}_{Y_ s \times Y_ s}$-module supported at $z$ whose value is $\kappa (z)$. Then $i_*\mathcal{O}_ z$ is the coherent $\mathcal{O}_{X \times _ S Y}$-module supported at $z$ whose value is $\kappa (z)$. Our assumption means that

\[ K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} i_*\mathcal{O}_ z = Li^*K \otimes _{\mathcal{O}_{Y_ s \times Y_ s}}^\mathbf {L} \mathcal{O}_ z = 0 \]

Hence by Lemma 57.11.3 we find an open neighbourhood $U(z) \subset X \times _ S Y$ of $z$ such that $K|_{U(z)} = 0$. In this case we set $Z(z) = \emptyset $ as closed subscheme of $U(z)$.

Case II: $z \in \Delta _{Y_ s/k}(Y_ s)$. Since $Y_ s$ is smooth over $k$ we know that $\Delta _{Y_ s/k} : Y_ s \to Y_ s \times Y_ s$ is a regular immersion, see More on Morphisms, Lemma 37.62.18. Choose a regular sequence $\overline{f}_1, \ldots , \overline{f}_ r \in \mathcal{O}_{Y_ s \times Y_ s, z}$ cutting out the ideal sheaf of $\Delta _{Y_ s/k}(Y_ s)$. Since a regular sequence is Koszul-regular (More on Algebra, Lemma 15.30.2) our assumption means that

\[ K_ z \otimes _{\mathcal{O}_{X \times _ S Y, z}}^\mathbf {L} \mathcal{O}_{Y_ s \times Y_ s, z} \in D(\mathcal{O}_{Y_ s \times Y_ s, z}) \]

is represented by the Koszul complex on $\overline{f}_1, \ldots , \overline{f}_ r$ over $\mathcal{O}_{Y_ s \times Y_ s, z}$. By Lemma 57.16.1 applied to $\mathcal{O}_{S, s} \to \mathcal{O}_{X \times _ S Y, z}$ we conclude that $K_ z \in D(\mathcal{O}_{X \times _ S Y, z})$ is represented by the Koszul complex on a regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_{X \times _ S Y, z}$ lifting the regular sequence $\overline{f}_1, \ldots , \overline{f}_ r$ such that moreover $\mathcal{O}_{X \times _ S Y}/(f_1, \ldots , f_ r)$ is flat over $\mathcal{O}_{S, s}$. By some limit arguments (Lemma 57.16.2) we conclude that there exists an affine open neighbourhood $U(z) \subset X \times _ S Y$ of $z$ and a closed subscheme $Z(z) \subset U(z)$ such that

$Z(z) \to U(z)$ is a regular closed immersion,

$K|_{U(z)}$ is quasi-isomorphic to $\mathcal{O}_{Z(z)}$,

$Z(z) \to S$ is flat,

$Z(z)_ s = \Delta _{Y_ s/k}(Y_ s) \cap U(z)_ s$ as closed subschemes of $U(z)_ s$.

By property (2), for $z, z' \in Y_ s \times Y_ s$, we find that $Z(z) \cap U(z') = Z(z') \cap U(z)$ as closed subschemes. Hence we obtain an open neighbourhood

\[ U = \bigcup \nolimits _{z \in Y_ s \times Y_ s\text{ closed}} U(z) \]

of $Y_ s \times Y_ s$ in $X \times _ S Y$ and a closed subscheme $Z \subset U$ such that (1) $Z \to U$ is a regular closed immersion, (2) $Z \to S$ is flat, and (3) $Z_ s = \Delta _{Y_ s/k}(Y_ s)$. Since $X \times _ S Y \to S$ is proper, after replacing $S$ by an open neighbourhood of $s$ we may assume $U = X \times _ S Y$. Since the projections $Z_ s \to Y_ s$ and $Z_ s \to X_ s$ are isomorphisms, we conclude that after shrinking $S$ we may assume $Z \to Y$ and $Z \to X$ are isomorphisms, see Lemma 57.16.3. This finishes the proof.
$\square$

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