The Stacks project

Lemma 57.18.5. Let $K$ be an algebraically closed field. There exists a countable set $I$ and for $i \in I$ a pair $(S_ i/K, X_ i \to S_ i, Y_ i \to S_ i, M_ i)$ with the following properties

  1. $S_ i$ is a scheme of finite type over $K$,

  2. $X_ i \to S_ i$ and $Y_ i \to S_ i$ are proper smooth morphisms of schemes,

  3. $M_ i \in D_{perf}(\mathcal{O}_{X_ i \times _{S_ i} Y_ i})$ is the Fourier-Mukai kernel of a relative equivalence from $X_ i$ to $Y_ i$ over $S_ i$, and

  4. for any smooth proper schemes $X$ and $Y$ over $K$ such that there is a $K$-linear exact equivalence $D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ there exists an $i \in I$ and a $s \in S_ i(K)$ such that $X \cong (X_ i)_ s$ and $Y \cong (Y_ i)_ s$.

Proof. Choose a countable subfield $k \subset K$ for example the prime field. By Lemmas 57.18.1 and 57.18.3 there exists a countable set of isomorphism classes of systems over $k$ satisfying parts (1), (2), (3) of the lemma. Thus we can choose a countable set $I$ and for each $i \in I$ such a system

\[ (S_{0, i}/k, X_{0, i} \to S_{0, i}, Y_{0, i} \to S_{0, i}, M_{0, i}) \]

over $k$ such that each isomorphism class occurs at least once. Denote $(S_ i/K, X_ i \to S_ i, Y_ i \to S_ i, M_ i)$ the base change of the displayed system to $K$. This system has properties (1), (2), (3), see Lemma 57.16.3. Let us prove property (4).

Consider smooth proper schemes $X$ and $Y$ over $K$ such that there is a $K$-linear exact equivalence $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$. By Proposition 57.14.4 we may assume that there exists an object $M \in D_{perf}(\mathcal{O}_{X \times Y})$ such that $F = \Phi _ M$ is the corresponding Fourier-Mukai functor. By Lemma 57.9.9 there is an $M'$ in $D_{perf}(\mathcal{O}_{Y \times X})$ such that $\Phi _{M'}$ is the right adjoint to $\Phi _ M$. Since $\Phi _ M$ is an equivalence, this means that $\Phi _{M'}$ is the quasi-inverse to $\Phi _ M$. By Lemma 57.9.9 we see that the Fourier-Mukai functors defined by the objects

\[ A = R\text{pr}_{13, *}( L\text{pr}_{12}^*M \otimes _{\mathcal{O}_{X \times Y \times X}}^\mathbf {L} L\text{pr}_{23}^*M') \]

in $D_{perf}(\mathcal{O}_{X \times X})$ and

\[ B = R\text{pr}_{13, *}( L\text{pr}_{12}^*M' \otimes _{\mathcal{O}_{Y \times X \times Y}}^\mathbf {L} L\text{pr}_{23}^*M) \]

in $D_{perf}(\mathcal{O}_{Y \times Y})$ are isomorphic to $\text{id} : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ X)$ and $\text{id} : D_{perf}(\mathcal{O}_ Y) \to D_{perf}(\mathcal{O}_ Y)$ Hence $A \cong \Delta _{X/K, *}\mathcal{O}_ X$ and $B \cong \Delta _{Y/K, *}\mathcal{O}_ Y$ by Lemma 57.14.5. Hence we see that $M$ is the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $K$ by definition.

We can write $K$ as the filtered colimit of its finite type $k$-subalgebras $A \subset K$. By Limits, Lemma 32.10.1 we can find $X_0, Y_0$ of finite type over $A$ whose base changes to $K$ produces $X$ and $Y$. By Limits, Lemmas 32.13.1 and 32.8.9 after enlarging $A$ we may assume $X_0$ and $Y_0$ are smooth and proper over $A$. By Lemma 57.16.4 after enlarging $A$ we may assume $M$ is the pullback of some $M_0 \in D_{perf}(\mathcal{O}_{X_0 \times _{\mathop{\mathrm{Spec}}(A)} Y_0})$ which is the Fourier-Mukai kernel of a relative equivalence from $X_0$ to $Y_0$ over $\mathop{\mathrm{Spec}}(A)$. Thus we see that $(S_0/k, X_0 \to S_0, Y_0 \to S_0, M_0)$ is isomorphic to $(S_{0, i}/k, X_{0, i} \to S_{0, i}, Y_{0, i} \to S_{0, i}, M_{0, i})$ for some $i \in I$. Since $S_ i = S_{0, i} \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(K)$ we conclude that (4) is true with $s : \mathop{\mathrm{Spec}}(K) \to S_ i$ induced by the morphism $\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(A) \cong S_{0, i}$ we get from $A \subset K$. $\square$


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