2016/04/29 by Vadim Vologodsky, Vologodsky, Vadim · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1604.08662
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 Zp with the special fiber i: X\hookrightarrow Y, we consider the functor L i^* ∘ 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 P1 then L i^* ∘ i_* is not of the Fourier-Mukai type. Note that by a theorem of Toen (\citet, Theorem 8.15) the latter assertion is equivalent to saying that L i^* ∘ i_* does not admit a lifting to a \mathbb Fp-linear DG quasi-functor Dbdg(X) → Dbdg(X), where Dbdg(X) is a (unique) DG enhancement of Db(X). However, essentially by definition, L i^* ∘ i_* lifts to a \mathbb Zp-linear DG quasi-functor.