Yoga for Fourier--Mukai partnership

TL;DR

This paper develops a framework for analyzing integral transforms under base change, extending existing results to singular varieties and arbitrary base fields, with applications to fibrations and singularities in arithmetic geometry.

Contribution

It generalizes Orlov's result to singular varieties and arbitrary base fields, providing a new 'yoga' of local algebra and fibers for testing derived equivalences.

Findings

Extended the theory of integral transforms to singular varieties.

Allowed arbitrary base fields in the analysis of derived equivalences.

Provided new insights into fibrations and singularities in arithmetic geometry.

Abstract

We study the behavior of integral transforms under base change. In particular, we establish a yoga of local algebra and fibers to test for derived equivalences or fully faithfulness via integral transforms. This generalizes a result of Orlov to singular varieties and strengthens several results in the literature by allowing arbitrary base fields. Additionally, it provides new insight into fibrations and their singularities in arithmetic settings (e.g.\ projective and flat schemes over a DVR).