Definition 57.15.1. Let S be a scheme. Let X \to S and Y \to S be smooth proper morphisms. An object K \in D_{perf}(\mathcal{O}_{X \times _ S Y}) is said to be the Fourier-Mukai kernel of a relative equivalence from X to Y over S if there exist an object K' \in D_{perf}(\mathcal{O}_{X \times _ S Y}) such that
\Delta _{X/S, *}\mathcal{O}_ X \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S X}}^\mathbf {L} L\text{pr}_{23}^*K')
in D(\mathcal{O}_{X \times _ S X}) and
\Delta _{Y/S, *}\mathcal{O}_ Y \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K' \otimes _{\mathcal{O}_{Y \times _ S X \times _ S Y}}^\mathbf {L} L\text{pr}_{23}^*K)
in D(\mathcal{O}_{Y \times _ S Y}). In other words, the isomorphism class of K defines an invertible arrow in the category defined in Section 57.14.
Comments (0)