Fourier-Mukai transform in the quantized setting

TL;DR

This paper establishes a criterion for equivalence of derived categories of DQ-modules based on the associated graded commutative kernels, extending the Fourier-Mukai transform to the quantized setting.

Contribution

It proves that a coherent DQ-kernel induces an equivalence if and only if its graded commutative kernel does, linking quantized and classical derived categories.

Findings

Equivalence of DQ-module categories is characterized by the associated graded kernels.

The result extends Fourier-Mukai transform principles to quantized algebraic geometry.

Provides a criterion for derived category equivalences in the quantized setting.

Abstract

We prove that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves.