On adjunctions for Fourier–Mukai transforms

We show that the adjunction counits of a Fourier–Mukai transform Φ:D(X1)→D(X2)Φ:D(X1)→D(X2) 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 ΦΦ. We also give another description of these maps, better suited to computing cones if the kernel of ΦΦ is a pushforward from a closed subscheme Z⊂X1×X2Z⊂X1×X2. Moreover, we show that we can replace the condition of properness of the ambient spaces X1X1 and X2X2 by that of ZZ 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.