Reflexive Modules, the Infinite Root Algebra and the Generating Hypothesis

TL;DR

This thesis explores the algebraic implications of Freyd's Generating Hypothesis, focusing on the construction of a self-injective ring with properties similar to the stable homotopy ring, using the infinite root algebra and Hahn series.

Contribution

It introduces a purely algebraic construction of a self-injective ring that models properties of the stable homotopy ring related to Freyd's hypothesis.

Findings

Proved that Theta-reflexive A-modules and multibasic A-modules are equivalent.

Established foundational results for the infinite root algebra and Hahn ring A.

Explored algebraic structures that mirror properties conjectured by Freyd's Generating Hypothesis.

Abstract

This thesis concerns the algebraic consequences of Freyd's Generating Hypothesis, and explores the question of whether there exists a self-injective ring R that can be constructed purely algebraically that exhibits some of the known properties of the stable homotopy ring, including some conjectured properties that follow from Freyd's Generating Hypothesis. As an example, we investigate the infinite root algebra of Hahn series P, firstly by establishing results for the related Hahn ring A. In particular, we prove that the Theta-reflexive A-modules and the multibasic A-modules are the same.