Triangulated Endofunctors of the Derived Category of Coherent Sheaves Which Do Not Admit DG Liftings

Abstract

In, Rizzardo and Van den Bergh (An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves. arXiv:1410.4039 , 2014) constructed an example of a triangulated functor between the derived categories of coherent sheaves on smooth projective varieties over a field k of characteristic 0 which is not of the Fourier–Mukai type. The purpose of this note is to show that if $${{\,\mathrm{{char}}\,}}k =p$$ then there are very simple examples of such functors. Namely, for a smooth projective Y over $${{\mathbb {Z}}}_p$$ with the special fiber $$i: X\hookrightarrow Y$$ , we consider the functor $$L i^* \circ i_*: D^b(X) \rightarrow D^b(X)$$ from the derived categories of coherent sheaves on X to itself. We show that if Y is a flag variety which is not isomorphic to $${{\mathbb {P}}}^1$$ then $$L i^* \circ i_*$$ is not of the Fourier–Mukai type. Note that by a theorem of Toen (Invent Math 167:615–667, 2007, Theorem 8.15) the latter assertion is equivalent to saying that $$L i^* \circ i_*$$ does not admit a lifting to a $${{\mathbb {F}}}_p$$ -linear DG quasi-functor $$D^b_{dg}(X) \rightarrow D^b_{dg}(X)$$ , where $$D^b_{dg}(X)$$ is a (unique) DG enhancement of $$D^b(X)$$ . However, essentially by definition, $$L i^* \circ i_*$$ lifts to a $${{\mathbb {Z}}}_p$$ -linear DG quasi-functor.