An elementary proof of a criterion for subfunctors of Ext to be closed

TL;DR

This paper provides an elementary proof that a subfunctor of Ext is closed if and only if it satisfies the 3x3-lemma property, avoiding complex embedding theorems and using basic abelian category techniques.

Contribution

It offers a self-contained, elementary proof of Buan's criterion for subfunctors of Ext to be closed, bypassing the need for advanced embedding theorems.

Findings

Elementary proof of Buan's criterion

Subfunctors of Ext are closed iff they satisfy the 3x3-lemma

Provides a self-contained exposition of the theory

Abstract

Let $\mathcal{A}$ be an abelian category and let $F$ be a subbifunctor of the additive bifunctor $\text{Ext}_{\mathcal{A}}^{1}(-,-)\colon \mathcal{A}^{\text{op}}\times \mathcal{A}\to \mathsf{Ab}$. Buan proved in [4] that $F$ is closed if, and only if, $F$ has the $3\times 3$-lemma property, a certain diagrammatic property satisfied by the class of $F$-exact sequences. The proof of this result relies on the theory of exact categories and on the Freyd--Mitchell embedding theorem, a very well-known overpowered result. In this paper we provide a proof of Buan's result only by means of elementary methods in abelian categories. To achieve this we survey the required theory of subfunctors leading us to a self-contained exposition of this topic.