On adjunctions for Fourier-Mukai transforms

摘要

We show that the adjunction counits of a Fourier-Mukai transform ΦD(X _1)→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 ZX _1×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.