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In this paper we prove that any smooth projective variety of dimension $ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier-Mukai functor between smooth projective schemes.
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Orlovs famous representability theorem asserts that any fully faithful functor between the derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. This result has been extended by Lunts and Orlov to include functors from perfect complexes to quasi-coherent complexes. In this paper we show that the latter extension is false without the full faithfulness hypothesis. Our results are based on the properties of scalar extensions of derived categories, whose investigation was started by Pawel Sosna and the first author.
A theorem by Orlov states that any equivalence between the bounded derived categories of coherent sheaves of two smooth projective varieties, X and Y, is isomorphic to a Fourier-Mukai transform with kernel in the bounded derived category of coherent sheaves of the product XxY. In the case of an exact functor which is not necessarily fully faithful, we compute some sheaves that play the role of the cohomology sheaves of the kernel, and that are isomorphic to the latter whenever an isomorphism to a Fourier-Mukai transform exists. We then exhibit a class of functors that are not full or faithful and still satisfy the above result.
We show that the adjunction counits of a Fourier-Mukai transform $Phi$ 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 $Phi$. We also give another description of these maps, better suited to computing cones if the kernel of $Phi$ 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.
Given a Fourier-Mukai functor $Phi$ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to $Phi$, and also give explicit formulas for them. These formulas are simple and natural, and recover the usual formulas when the Fourier-Mukai kernel is a perfect complex. This extends previous work of Anno and Logvinenko, and Hernandez Ruiperez, Lopez Martin and Sancho de Salas, and has applications to the twist autoequivalences of Donovan and Wemyss.
Orlovs famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis.
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