Triangulated endofunctors of the derived category of coherent sheaves which do not admit DG liftings

Abstract

Recently, Rizzardo and Van den Bergh 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 $chark=p$ 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 $L{i}^{\ast }\circ i_{\ast }:D^b(X)\to D^b(X)$ 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 $L{i}^{\ast }\circ i_{\ast }$ is not of the Fourier-Mukai type. Note that by a theorem of Toen (\cite{t}, Theorem 8.15) the latter assertion is equivalent to saying that $L{i}^{\ast }\circ i_{\ast }$ does not admit a lifting to a ${\mathbb{F}}_p$-linear DG quasi-functor $D_{dg}^b(X)\to D_{dg}^b(X)$, where $D_{dg}^b(X)$ is a (unique) DG enhancement of $D^b(X)$. However, essentially by definition, $L{i}^{\ast }\circ i_{\ast }$ lifts to a ${\mathbb{Z}}_p$-linear DG quasi-functor.