Fourier–Mukai transform

Remarks

$\Phi_P$ 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. Example: 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 $\Phi_{\mathcal{O}_\Delta} \cong \id$. For 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. An example that is not an equivalence when $\dim Y > 0$: for $P = \mathcal{O}_{X \times Y}$, flat base change gives $\Phi_P(E) \cong R\Gamma(X, E) \otimes_k \mathcal{O}_Y$. So 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.