Failure of flat descent of model structures on module categories

TL;DR

This paper demonstrates that the collection of model structures on module categories does not satisfy flat descent, showing a fundamental limitation in the behavior of these structures over Artin stacks.

Contribution

It proves that the collection of model structures on module categories fails to obey flat descent, revealing a key obstruction in their geometric behavior.

Findings

Model structures on module categories do not obey flat descent.

The collection fails to be a separated presheaf in the fppf topology.

This failure impacts the understanding of module categories over Artin stacks.

Abstract

We prove that the collection of model structures on (quasicoherent) module categories does not obey flat descent. In particular, it fails to be a separated presheaf, in the fppf topology, on Artin stacks.