The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Journal of Noncommutative Geometry


Full-Text PDF (270 KB) | Metadata | Table of Contents | JNCG summary
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