On adjunctions for Fourier-Mukai transforms

Anno, Rina and Logvinenko, Timothy ORCID: https://orcid.org/0000-0001-5279-6977 2012. On adjunctions for Fourier-Mukai transforms. Advances in Mathematics 231 (3-4) , pp. 2069-2115. 10.1016/j.aim.2012.06.007

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Abstract

We show that the adjunction counits of a Fourier–Mukai transform Φ:D(X1)→D(X2) arise from maps of the kernels of the corresponding Fourier–Mukai transforms. In a very general setting of proper separable schemes of finite type over a field we write down these maps of kernels explicitly –facilitating the computation of the twist (the cone of an adjunction counit) of Φ. We also give another description of these maps, better suited to computing cones if the kernel of Φ is a pushforward from a closed subscheme Z⊂X1×X2. Moreover, we show that we can replace the condition of properness of the ambient spaces X1 and X2 by that of Z being proper over them and still have this description apply as is. This can be used, for instance, to compute spherical twists on non-proper varieties directly and in full generality.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	Fourier–Mukai transforms; Derived categories; Algebraic geometry; Spherical twists; Adjunction; Künneth map
Publisher:	Elsevier
ISSN:	0001-8708
Date of First Compliant Deposit:	30 March 2016
Last Modified:	09 May 2023 20:11
URI:	https://orca.cardiff.ac.uk/id/eprint/45323

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