# An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves

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## Abstract

Orlov’s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier–Mukai functor. In this paper we show that this result is false without the fully faithfulness hypothesis. We also show that our functor does not lift to the homotopy category of spectral categories if the ground field is \({\mathbb Q}\).

## Mathematics Subject Classification

13D09 18E30 14A22## Notes

### Acknowledgements

The authors thank Greg Stevenson and Adam-Christiaan Van Roosmalen for a number of interesting discussions which reinforced our interest in the problem.

Alice Rizzardo would like to thank the University of Hasselt for its hospitality during two recent visits to Belgium. She would also like to thank MSRI for providing a welcoming environment during the Noncommutative Geometry and Representation Theory semester, which encouraged many fruitful conversations, some of which got this project started. She is grateful to Jon Pridham for providing very useful ideas and guidance for “Appendix B”, and to Julian Holstein for help with understanding stable infinity categories. Finally, she would like to thank her advisor, Johan de Jong, for originally suggesting she look into exact functors between derived categories in the non fully faithful case.

Michel Van den Bergh thanks the International School for Advanced Studies (SISSA) in the beautiful city of Trieste for the wonderful working conditions it provides.

The authors thank Andrei Căldăraru for useful discussions concerning derived self intersections and Fourier–Mukai functors. They are deeply grateful to Wendy Lowen for allowing the use of some ideas developed during an ongoing cooperation with the second author.

Finally the authors thank Amnon Neeman for providing them with an alternative and much more general proof for the extension of (1.1).

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