- journal article metadata
European Mathematical Society Publishing House
2017-12-18 23:40:02
Journal of Noncommutative Geometry
J. Noncommut. Geom.
JNCG
1661-6952
1661-6960
Global analysis, analysis on manifolds
General
10.4171/JNCG
http://www.ems-ph.org/doi/10.4171/JNCG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
11
2017
4
Representability of cohomological functors over extension fields
Alice
Rizzardo
SISSA, Trieste, Italy
Representability, base extension, Fourier–Mukai
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 $Y$.
Category theory; homological algebra
Algebraic geometry
1267
1287
10.4171/JNCG/11-4-2
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-2
12
15
2017