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  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ and $Y$ be smooth [[projective varieties|A4H4CZ2V35]] over $k$.\nLet $D^b(X)$ be the [[bounded derived category of coherent sheaves|2VP2D51KB8]] on $X$, and let $p \\colon X \\times Y \\to X$ and $q \\colon X \\times Y \\to Y$ be the projections.\nFor an object $P$ of $D^b(X \\times Y)$, the *Fourier–Mukai transform* with kernel $P$ is the functor\n$$\\Phi_P \\colon D^b(X) \\to D^b(Y), \\qquad \\Phi_P(E) = Rq_*\\bigl(p^* E \\otimes^L P\\bigr).$$\nHere $p^*$ is the pullback, which is exact since $p$ is flat.\nThe symbols $\\otimes^L$ and $Rq_*$ denote the derived tensor product and the derived pushforward, which are [[derived functors|NPMC3K6J2F]].\n\n## Remarks\n\n$\\Phi_P$ is an exact functor of triangulated categories, and composites of Fourier–Mukai transforms are again Fourier–Mukai transforms, with kernel given by a convolution of the kernels.\nExample: for $Y = X$ and $P = \\mathcal{O}_\\Delta$, the structure sheaf of the diagonal $\\Delta \\subseteq X \\times X$, one has $p^* E \\otimes^L \\mathcal{O}_\\Delta \\cong \\delta_* E$ for the diagonal embedding $\\delta$, and $q \\circ \\delta = \\id_X$, so $\\Phi_{\\mathcal{O}_\\Delta} \\cong \\id$.\nFor an abelian variety $A$ with dual $\\hat A$ and Poincaré line bundle $\\mathcal{P}$ on $A \\times \\hat A$, Mukai proved that $\\Phi_{\\mathcal{P}} \\colon D^b(A) \\to D^b(\\hat A)$ is an equivalence.\nAn example that is not an equivalence when $\\dim Y \u003e 0$: for $P = \\mathcal{O}_{X \\times Y}$, flat base change gives $\\Phi_P(E) \\cong R\\Gamma(X, E) \\otimes_k \\mathcal{O}_Y$.\nSo every object goes to a sum of shifts of $\\mathcal{O}_Y$, and the structure sheaf of a closed point of $Y$ is not in the image.\n",
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