The upper topology and its relation with the projective modules

TL;DR

This paper explores the upper topology on ordinal numbers and establishes a connection between the continuity of the rank map of projective modules and their properties, revealing new insights into module theory.

Contribution

It introduces new results on the upper topology and demonstrates the equivalence between the continuity of the rank map and projectivity for finitely generated flat modules.

Findings

The rank map of a locally finite type projective module is continuous in the upper topology.

Finitely generated flat modules are projective if and only if their rank map is upper topology continuous.

The upper topology provides a natural setting for analyzing the continuity properties of module rank maps.

Abstract

In this paper, first we obtain some new and interesting results on projective modules and on the upper topology of an ordinal number. Then it is shown that the rank map of a locally of finite type projective module is continuous with respect to the upper topology (by contract, it is well known this map is not necessarily continuous with respect to the discrete topology). It is also proved that a finitely generated flat module is projective if and only if its rank map is continuous with respect to the upper topology.