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

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## 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.

## Notes

### Acknowledgements

I would like to thank Alberto Canonaco and Paolo Stellari: their interest prompted writing this note. Also, I am grateful to Alexander Samokhin for stimulating discussions and references. I would like to thanks the referee for his or her comments which helped to improve the exposition. The author was partially supported by the Laboratory of Mirror Symmetry NRU HSE, RF Government Grant, ag. N^{o} 14.641.31.0001.

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