On adjunctions for Fourier-Mukai transforms

We show that the adjunction counits of a Fourier-Mukai transform $Φ$ from $D(X_1)$ to $D(X_2)$ arise from maps of the kernels of the corresponding Fourier-Mukai transforms. In a very general setting of proper separable schemes of finite type over a field we write down these maps of kernels explicitly -- facilitating the computation of the twist (the cone of an adjunction counit) of $Φ$. We also give another description of these maps, better suited to computing cones if the kernel of $Φ$ is a pushforward from a closed subscheme $Z$ of $X_1 \times X_2$. Moreover, we show that we can replace the condition of properness of the ambient spaces $X_1$ and $X_2$ by that of $Z$ being proper over them and still have this description apply as is. This can be used, for instance, to compute spherical twists on non-proper varieties directly and in full generality.

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