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# Journal of Noncommutative Geometry

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**Volume 11, Issue 4, 2017, pp. 1267–1287**

**DOI: 10.4171/JNCG/11-4-2**

Published online: 2017-12-15

Representability of cohomological functors over extension fields

Alice Rizzardo^{[1]}(1) SISSA, Trieste, Italy

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

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

*Keywords: *Representability, base extension, Fourier–Mukai

Rizzardo Alice: Representability of cohomological functors over extension fields. *J. Noncommut. Geom.* 11 (2017), 1267-1287. doi: 10.4171/JNCG/11-4-2