Definition 57.8.1. Let S be a scheme. Let X and Y be schemes over S. Let K \in D(\mathcal{O}_{X \times _ S Y}). The exact functor
of triangulated categories is called a Fourier-Mukai functor and K is called a Fourier-Mukai kernel for this functor. Moreover,
if \Phi _ K sends D_\mathit{QCoh}(\mathcal{O}_ X) into D_\mathit{QCoh}(\mathcal{O}_ Y) then the resulting exact functor \Phi _ K : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y) is called a Fourier-Mukai functor,
if \Phi _ K sends D_{perf}(\mathcal{O}_ X) into D_{perf}(\mathcal{O}_ Y) then the resulting exact functor \Phi _ K : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y) is called a Fourier-Mukai functor, and
if X and Y are Noetherian and \Phi _ K sends D^ b_{\textit{Coh}}(\mathcal{O}_ X) into D^ b_{\textit{Coh}}(\mathcal{O}_ Y) then the resulting exact functor \Phi _ K : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ Y) is called a Fourier-Mukai functor. Similarly for D_{\textit{Coh}}, D^+_{\textit{Coh}}, D^-_{\textit{Coh}}.
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