On extensions of covariantly finite subcategories

TL;DR

This paper investigates the conditions under which covariantly finite subcategories are preserved under extensions, providing counterexamples in general abelian categories and establishing a triangulated category version of the theorem.

Contribution

It presents a counterexample to the generalization of Gentle-Todorov's theorem and proves a triangulated category analogue, with applications to classical and recent results.

Findings

Counterexample showing failure in arbitrary abelian categories

Triangulated version of Gentle-Todorov's theorem proven

Simplified proofs of classical and recent results

Abstract

In \cite{GT}, Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is still covariantly finite. We give an counterexample to show that Gentle-Todorov's theorem may fail in arbitrary abelian categories; we also prove that a triangulated version of Gentle-Todorov's theorem holds; we make applications of Gentle-Todorov's theorem to obtain short proofs to a classical result by Ringel and a recent result by Krause and Solberg.