Reflexivity of modules

TL;DR

The paper proves that for modules viewed as functors, the natural morphism to its double dual is an isomorphism, establishing a reflexivity property in this functorial context.

Contribution

It introduces a functorial perspective on modules and proves their reflexivity via an isomorphism to their double duals.

Findings

The functor of modules is reflexive under the double dual operation.

The natural morphism from a module to its double dual is an isomorphism.

This establishes a functorial reflexivity property for modules.

Abstract

We consider $\,R-$modules as functors in the following way: if $\,M\,$ is a (left) $R$-module, let $\,\mathcal M\,$ be the functor of $\,\mathcal R-$modules defined by $\,\mathcal M(S) := S \otimes_R M\,$ for every $\,R-$algebra $\,S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\,\mathcal M\to \mathcal M^{**}\,$ is an isomorphism.