Adjoints to a Fourier-Mukai transform

TL;DR

This paper derives explicit formulas for left and right adjoints of Fourier-Mukai functors on singular schemes, extending previous results and enabling applications to autoequivalences in algebraic geometry.

Contribution

It provides new explicit formulas for adjoints of Fourier-Mukai functors on singular schemes, generalizing prior work and broadening their applicability.

Findings

Explicit formulas for adjoints are simple and natural.

Formulas recover classical cases when the kernel is perfect.

Applications include twist autoequivalences in derived categories.

Abstract

Given a Fourier-Mukai functor $\Phi$ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to $\Phi$, and also give explicit formulas for them. These formulas are simple and natural, and recover the usual formulas when the Fourier-Mukai kernel is a perfect complex. This extends previous work of Anno and Logvinenko, and Hernandez Ruiperez, Lopez Martin and Sancho de Salas, and has applications to the twist autoequivalences of Donovan and Wemyss.