Characterization of quasi-coherent modules that are module schemes

TL;DR

This paper investigates when quasi-coherent modules over a ring are actually module schemes, linking this to the characterization of finite type projective modules, thus advancing the understanding of module representations.

Contribution

It provides a characterization of when quasi-coherent modules are module schemes, connecting this to finite type projective modules.

Findings

Quasi-coherent modules are module schemes if and only if they correspond to finite type projective modules.

Establishes a criterion for identifying module schemes among quasi-coherent modules.

Enhances the theoretical framework for linear representations of affine group schemes.

Abstract

The R-module functors that are essential for the development of the theory of the linear representations of an affine R-group are the quasi-coherent R-modules and the R-module schemes. The aim of this paper is to study when a quasi-coherent R-module is an R-module scheme. We will prove that it is equivalent to giving a characterization of projective R-modules of finite type.