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

TL;DR

This paper demonstrates simple examples of triangulated functors in positive characteristic that are not of Fourier-Mukai type and do not admit DG liftings, contrasting with characteristic zero cases.

Contribution

It provides explicit examples of such functors in characteristic p, showing they differ from characteristic zero cases and do not admit DG liftings.

Findings

For certain flag varieties, the functor is not Fourier-Mukai.

Such functors do not admit liftings to DG quasi-functors over F_p.

These examples contrast with characteristic zero cases where liftings exist.

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 $char \, k =p$ 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 $L i^* \circ i_*: D^b(X) \to D^b(X)$ 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 $L i^* \circ i_*$ 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 $L i^* \circ i_*$ does not admit a lifting to a $\mathbb F_p$-linear DG quasi-functor $D^b_{dg}(X) \to D^b_{dg}(X)$, 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.