LAUSR

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 $${{\,\mathrm{{char}}\,}}k =p$$chark=p then there are very simple examples of such functors. Namely, for a smooth projective Y over $${{\mathbb {Z}}}_p$$Zp with the special fiber $$i: X\hookrightarrow Y$$i:X↪Y, we consider the functor $$L i^* \circ i_*: D^b(X) \rightarrow D^b(X)$$Li∗∘i∗:Db(X)→Db(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 $${{\mathbb {P}}}^1$$P1 then $$L i^* \circ i_*$$Li∗∘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 $$L i^* \circ i_*$$Li∗∘i∗ does not admit a lifting to a $${{\mathbb {F}}}_p$$Fp-linear DG quasi-functor $$D^b_{dg}(X) \rightarrow D^b_{dg}(X)$$Ddgb(X)→Ddgb(X), where $$D^b_{dg}(X)$$Ddgb(X) is a (unique) DG enhancement of $$D^b(X)$$Db(X). However, essentially by definition, $$L i^* \circ i_*$$Li∗∘i∗ lifts to a $${{\mathbb {Z}}}_p$$Zp-linear DG quasi-functor.