Flat relative Mittag-Leffler modules and Zariski locality

TL;DR

This paper investigates the ascent and descent properties of relative Mittag-Leffler modules, establishing Zariski locality of certain classes of quasi-coherent sheaves and vector bundles across schemes.

Contribution

It extends the theory of relative Mittag-Leffler modules by proving Zariski locality of locally f-projective sheaves and n-Drinfeld vector bundles on schemes.

Findings

Zariski locality of locally f-projective quasi-coherent sheaves for all schemes.

Zariski locality of n-Drinfeld vector bundles on locally noetherian schemes.

Extension of ascent and descent properties for relative Mittag-Leffler modules.

Abstract

The ascent and descent of the Mittag-Leffler property were instrumental in proving Zariski locality of the notion of an (infinite dimensional) vector bundle by Raynaud and Gruson in \cite{RG}. More recently, relative Mittag-Leffler modules were employed in the theory of (infinitely generated) tilting modules and the associated quasi-coherent sheaves, \cite{AH}, \cite{HST}. Here, we study the ascent and descent along flat and faithfully flat homomorphisms for relative versions of the Mittag-Leffler property. In particular, we prove the Zariski locality of the notion of a locally f-projective quasi-coherent sheaf for all schemes, and for each $n \geq 1$, of the notion of an $n$-Drinfeld vector bundle for all locally noetherian schemes.