Flat relative Mittag-Leffler modules and Zariski locality

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 $n\ge 1$, of the notion of an $n$-Drinfeld vector bundle for all locally noetherian schemes.

arXiv commentRevised version, extending the main results of v1 from classes of finite type to definable closures of Tor-orthogonal classes. The latter have more applications, e.g., to n-Drinfeld vector bundles

References in corpus (6)

• Tilting, cotilting, and spectra of commutative noetherian rings
• Tilting classes over commutative rings
• Zariski locality of quasi-coherent sheaves associated with tilting
• On flat generators and Matlis duality for quasicoherent sheaves
• Flat relative Mittag-Leffler modules and approximations
• Flat Mittag-Leffler modules, and their relative and restricted versions