Virtual Extensions of Modules

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.