Walker's cancellation theorem

TL;DR

This paper investigates Walker's cancellation theorem in the context of diagram categories of abelian groups, providing a counterexample that shows the theorem fails constructively and contrasting it with cases where the endomorphism ring has stable range one.

Contribution

The paper constructs a specific example demonstrating the failure of Walker's cancellation theorem in diagram categories, highlighting limitations of constructive proofs in this setting.

Findings

Counterexample in diagram category where the theorem fails

Constructive proof is not possible in the general case

Stable range one endomorphism rings allow constructive proofs

Abstract

Walker's cancellation theorem says that if B+Z is isomorphic to C+Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof even if B and C are subgroups of the free abelian group on two generators. Both of these results contrast with a group whose endomorphism ring has stable range one, which allows a constructive proof of cancellation and also a proof in any diagram category.