Zariski locality of quasi-coherent sheaves associated with tilting

TL;DR

This paper extends the Zariski locality property from vector bundles to a broader class of quasi-coherent sheaves associated with n-tilting modules on all schemes, and explores descent properties of tilting modules.

Contribution

It generalizes the Zariski locality of vector bundles to all n-tilting sheaves and establishes descent results for tilting modules over various ring morphisms.

Findings

Zariski locality holds for all n-tilting quasi-coherent sheaves.

Tilting modules descend along faithfully flat ring morphisms in key cases.

The results unify and extend classical locality results for vector bundles.

Abstract

A classic result by Raynaud and Gruson says that the notion of an (infinite dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for n = 0) of the locality of more general notions of quasi-coherent sheaves related to (infinite dimensional) n-tilting modules and classes. Here, we prove the latter locality for all n and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is noetherian).