Yoga for Fourier--Mukai partnership
Pat Lank, Kabeer Manali-Rahul, Nebojsa Pavic
TL;DR
This work advances the study of Fourier–Mukai type integral transforms under base change, especially over singular varieties and arbitrary base fields. By developing a base-change framework anchored in local algebra and fiber analysis, the authors establish ascent/descent results for adjunctions, full faithfulness, and equivalences, extending Orlov’s results to broader contexts. They prove that adjointness and derived-equivalence properties survive affine base changes (under suitable tor-dimension hypotheses) and derive arithmetic-geometric consequences, including invariance of genus, smoothness, and CM/Gorenstein properties under FM partnership. The results illuminate how derived categories encode geometric and arithmetic invariants across base changes and provide tools for identifying k-forms and base-change isomorphisms for curves and fibrations.
Abstract
We study the behavior of integral transforms under various forms of base change. In particular, we establish a yoga of local algebra and fibers to test for derived equivalences via integral transforms. This partially generalizes a result of Orlov to singular varieties and strengthens many results in the literature to arbitrary fields. Additionally, it provides new insight into fibrations and their singularities arising in more arithmetically flavored contexts (e.g.\ projective and flat schemes over a DVR).