On adjunctions for Fourier–Mukai transforms

Abstract

We show that the adjunction counits of a Fourier-Mukai transform Φ : D(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 Z ⊂ X 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.On the other hand, the identity functor Id is the Fourier-Mukai transform D(X 1 ) → D(X 1 ) with kernelConsider now the left adjunction counitIn general, morphisms between Fourier-Mukai kernels map neither injectively nor surjectively to natural transformations between the Fourier-Mukai transforms. Thus there is no a priori reason for (1.4) to come 1 arXiv:1004.3052v3 [math.AG]

Orthogonally spherical objects and spherical fibrations

“…We need this to compute F E since co-twist functors need to be defined as the cones of adjunction units. We get this result for free from the similar result for adjunction co-units in [AL12] using the Grothendieck duality arguments summarized in §2.1. We then review the formalism of spherical functors in Section §2.3.…”

“…Moreover, to compute the twist T S and the co-twist F S of S we need to write down the units and the co-units of these adjunctions on the level of Fourier-Mukai kernels. Partly this was achieved in §3.1 of [AL12]. We give a brief summary here.…”

“…If X 1 and X 2 are not proper, but the support of E is proper over X 1 and X 2 , one can still apply the above results via compactification as described in §3.2 of [AL12]. If E is a pushforward of an object in the derived category of a closed subscheme W ֒→ X 1 × X 2 proper over both X 1 and X 2 one can also dualize Theorems 4.1 and 4.2 of [AL12] to obtain an alternative description of morphisms of kernels which induce both adjunction units.…”

“…If E is a pushforward of an object in the derived category of a closed subscheme W ֒→ X 1 × X 2 proper over both X 1 and X 2 one can also dualize Theorems 4.1 and 4.2 of [AL12] to obtain an alternative description of morphisms of kernels which induce both adjunction units. We leave this as an exercise for the reader.…”

“…However in [AL12] it is shown that in a very general context we can represent both functors in (1.1) by Fourier-Mukai kernels and then represent the adjunction co-unit (1.1) by a natural morphism µ between these kernels. We can therefore define the twist functor T E as the Fourier-Mukai transform whose kernel is the cone of µ and consider the following:…”