Virtual Extensions of Modules

TL;DR

This paper introduces the concept of virtual categories and diagrams for modules, providing a new perspective on extensions and subquotients, and extends classical correspondences to a virtual setting.

Contribution

It develops the theory of virtual categories and diagrams for modules, including the virtuality group, and generalizes the Yoneda correspondence in this new framework.

Findings

Introduces the virtual category of a module with objects as submodules of subquotients.

Defines the virtuality group A(D) generated by virtual constituents of a module.

Upgrades the Yoneda correspondence to a bimodule isomorphism in the virtual context.

Abstract

In this article we are examining extensions and some basic diagrammatic properties of modules, in both cases from a new, "virtual" point of view. As natural background for investigating the kind of problems we are dealing with, the virtual category of a module M is introduced, having as objects the submodules of M's subquotients modulo some identifications. In the case of extensions our approach implies viewing "proportionality classes" of extensions of (dually, by) a simple module by (resp. of) another simple as quotients of a certain quotient (which is in fact a subdirect product) of a projective cover, that comprises all those classes - or dually as submodules of a comprising submodule (which is a push-out) of an injective hull. In particular we become thus able to upgrade the Yoneda correspondence to a bimodule isomorphism. Basic steps toward the foundation of and investigation into the theory of Virtual Diagrams are also made here. In particular, the "virtuality group" A(D) of a virtual diagram D of a module M is introduced, generated by the D-visible virtual constituents of M, with respect to an addition that generalizes the one of submodules in a module.