Descent of restricted flat Mittag-Leffler modules and generalized vector bundles

TL;DR

This paper proves that the property of being a restricted flat Mittag-Leffler module or a generalized vector bundle is local on schemes, extending previous results to all infinite cardinals using advanced filtration techniques.

Contribution

It establishes the locality of restricted flat Mittag-Leffler modules and generalized vector bundles for all infinite cardinals, using refined filtration methods instead of traditional de9vissage.

Findings

Confirmed locality for all infinite cardinal restrictions.

Extended previous results to a broader class of modules.

Applied Hill Lemma and c7-filtrations for the proof.

Abstract

A basic question for any property of quasi--coherent sheaves on a scheme $X$ is whether the property is local, that is, it can be defined using any open affine covering of $X$. Locality follows from the descent of the corresponding module property: for (infinite dimensional) vector bundles and Drinfeld vector bundles, it was proved by Kaplansky's technique of d\'evissage already in \cite[II.\S3]{RG}. Since vector bundles coincide with $\aleph_0$-restricted Drinfeld vector bundles, a question arose in \cite{EGPT} of whether locality holds for $\kappa$-restricted Drinfeld vector bundles for each infinite cardinal $\kappa$. We give a positive answer here by replacing the d\' evissage with its recent refinement involving $\mathcal C$-filtrations and the Hill Lemma.