Flat relative Mittag-Leffler modules and Zariski locality
Asmae Ben Yassine, Jan Trlifaj
TL;DR
The paper addresses the problem of establishing Zariski locality for quasi-coherent sheaves attached to relative Mittag-Leffler notions in broad scheme-theoretic contexts, including locally noetherian settings. It develops ascent/descent theory for flat relative Mittag-Leffler modules, leveraging definable dualities and pure-monomorphism techniques to transfer properties across flat base changes and to descend along faithfully flat maps. The main contributions are the Zariski locality results for locally $f$-projective sheaves on all schemes and for $n$-Drinfeld vector bundles on locally noetherian schemes, together with a robust algebraic framework for relative Mittag-Leffler modules that clarifies behavior under base change and finite-type dualities. These results generalize classical Raynaud–Gruson locality, enriching the study of infinite-dimensional vector bundles and tilting-theory-related quasi-coherent sheaves across arbitrary schemes, with potential implications for Drinfeld vector bundles and tilting/silting phenomena.
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.