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

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 previously proved by Kaplansky’s technique of dévissage. Since vector bundles coincide with $\aleph _0$–restricted Drinfeld vector bundles, a question arose as to whether locality holds for $\kappa$–restricted Drinfeld vector bundles for each infinite cardinal $\kappa$. We give a positive answer here by replacing the dévissage with its recent refinement involving $\mathcal C$–filtrations and the Hill Lemma.