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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