On adjunctions for Fourier-Mukai transforms

TL;DR

This paper explores the structure of adjunctions in Fourier-Mukai transforms, providing explicit kernel maps and methods to compute twists, especially for non-proper varieties, enhancing understanding and computational techniques in derived categories.

Contribution

It introduces explicit kernel maps for adjunction counits of Fourier-Mukai transforms and extends the framework to non-proper schemes, facilitating the computation of twists and spherical twists.

Findings

Explicit kernel maps for adjunction counits are derived.

The framework applies to non-proper varieties via properness over a subscheme.

Methods for computing twists and spherical twists are developed.

Abstract

We show that the adjunction counits of a Fourier-Mukai transform $\Phi$ from $D(X_1)$ to $D(X_2)$ 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 $\Phi$. We also give another description of these maps, better suited to computing cones if the kernel of $\Phi$ is a pushforward from a closed subscheme $Z$ of $X_1 \times X_2$. Moreover, we show that we can replace the condition of properness of the ambient spaces $X_1$ and $X_2$ 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.