Countability of relative Fourier-Mukai partners

TL;DR

This paper extends the countability result of Fourier-Mukai partners from complex varieties to smooth proper schemes over arbitrary noetherian bases, broadening the understanding of derived equivalences in algebraic geometry.

Contribution

It generalizes the countability of Fourier-Mukai partners to smooth proper schemes over noetherian bases, building on previous results for complex varieties.

Findings

Countably many Fourier-Mukai partners over noetherian bases

Extension of known results from complex to arbitrary algebraically closed fields

Upgrade of proof techniques for broader class of schemes

Abstract

Anel and To\"en proved that a smooth projective complex variety has only countably many smooth projective Fourier-Mukai partners up to isomorphism. This is generalized in the Stacks Project to the case where the varieties are smooth proper over an arbitrary algebraically closed field. This note will upgrade the proof of the latter reference to show that a smooth proper scheme over a noetherian base has only countably many relative Fourier-Mukai partners up to isomorphism.