Representability of cohomological functors over extension fields

Abstract

We generalize a result of Orlov and Van den Bergh on the representability of a cohomological functor $H:D_{\mathrm{Coh}}^b(X)\to {\underline{\mathrm{mod}}}_L$ to the case where $L$ is a field extension of the base field $k$ of the variety $X$, with trdeg${}_{k}L\le 1$ or $L$ purely transcendental of degree 2.

This result can be applied to investigate the behavior of an exact functor $F:D_{\mathrm{Coh}}^b(X)\to D_{\mathrm{Coh}}^b(Y)$ with $X$ and $Y$ smooth projective varieties and dim $Y\le 1$ or $Y$ a rational surface. We show that for any such $F$ there exists a "generic kernel" $A$ in $D_{\mathrm{Coh}}^b(X\times Y)$, such that $F$ is isomorphic to the Fourier–Mukai transform with kernel $A$ after composing both with the pullback to the generic point of $Y$.