arXiv subjects

Asmae Ben Yassine

Publications and source records attributed to Asmae Ben Yassine.

3 recordsLinked to original sources

arXiv2024

Dualizations of approximations, $\aleph_1$-projectivity, and Vopěnka's Principles

The approximation classes of modules that arise as components of cotorsion pairs are tied up by Salce's duality. Here we consider general approximation classes of modules and investigate possibilities of dualization in dependence on closure properties of these classes. While some proofs are easily dualized, other dualizations require large cardinal principles, and some fail in ZFC, with counterexamples provided by classes of $\aleph_1$-projective modules over non-perfect rings. For example, we show that Vopěnka's Principle implies that each covering class of modules closed under homomorphic images is of the form Gen($M$) for a module $M$, and that the latter property restricted to classes generated by $\aleph_1$-free abelian groups implies Weak Vopěnka's Principle.

math.RT↗

arXiv2023

Flat relative Mittag-Leffler modules and Zariski locality

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.

math.AG↗

arXiv2021

Flat relative Mittag-Leffler modules and approximations

The classes $\mathcal D _{\mathcal Q}$ of flat relative Mittag-Leffler modules are sandwiched between the class $\mathcal F \mathcal M$ of all flat (absolute) Mittag-Leffler modules, and the class $\mathcal F$ of all flat modules. Building on the works of Angeleri H\" ugel, Herbera, and \v Saroch, we give a characterization of flat relative Mittag-Leffler modules in terms of their local structure, and show that Enochs' Conjecture holds for all the classes $\mathcal D _{\mathcal Q}$. In the final section, we apply these results to the particular setting of f-projective modules.

math.RT↗