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, 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 Toën (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.