Zariski Locality of Quasi-Coherent Sheaves Associated with Tilting

Abstrakt

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).

Klíčová slova

• Tilting module
• locally tilting quasi-coherent sheaf
• Zariski locality
• Mittag-Leffler module