Integral transforms on singularity categories for Noetherian schemes

TL;DR

This paper characterizes when integral transforms induce exact functors on singularity categories of schemes, extending previous work and analyzing adjoints, with implications for derived equivalences of singular varieties.

Contribution

It provides a complete characterization of integral transforms on singularity categories for schemes proper over a Noetherian base, extending earlier results and analyzing adjoint functors.

Findings

Complete characterization of integral transforms inducing exact functors

Extension of Olander's result to varieties with mild singularities

Obstruction identified for derived equivalences between singular varieties

Abstract

This work studies conditions under which integral transforms induce exact functors on singularity categories between schemes that are proper over a Noetherian base scheme. A complete characterization for this behavior is provided, which extends earlier work of Ballard and Rizzardo. We leverage a description of the bounded derived category of coherent sheaves as finite cohomological functors on the category of perfect complexes, which is an application of Neeman's approximable triangulated categories, to reduce arguments to an affine local setting. Moreover, we study adjoints of such functors, extend a result of Olander to varieties with mild singularities, and provide an obstruction for derived equivalences between singular varieties.