Ideal approximation theory in Frobenius categories

TL;DR

This paper investigates the relationships between special precovering ideals in Frobenius categories and their stable categories, establishing conditions for completeness of ideal cotorsion pairs and extending classical lemmas to this setting.

Contribution

It introduces an ideal version of the Bongartz-Eklof-Trlifaj Lemma and characterizes the completeness of cotorsion pairs in Frobenius categories and their stable categories.

Findings

Precovering ideals with identity on projectives are special.

Complete cotorsion pairs correspond to precovering or preenveloping ideals.

An ideal version of the Bongartz-Eklof-Trlifaj Lemma is established.

Abstract

Let $\mathcal{A}$ be a Frobenius category and $\omega$ the full subcategory consisting of projective objects. The relations between special precovering (resp., precovering) ideals in $\mathcal{A}$ and special precovering (resp., preenveloping) ideals in the stable category $\mathcal{A}/\omega$ are explored. In combination with a result due to Breaz and Modoi, we conclude that every precovering or preenveloping ideal $\mathcal{I}$ in $\mathcal{A}$ with $1_{X}\in{\mathcal{I}}$ for any $X\in{\omega}$ is special. As a consequence, it is proved that an ideal cotorsion pair $(\mathcal{I},\mathcal{J})$ in $\mathcal{A}$ is complete if and only if $\mathcal{I}$ is precovering if and only if $\mathcal{J}$ is preenveloping. This leads to an ideal version of the Bongartz-Eklof-Trlifaj Lemma in $\mathcal{A}/\omega$, which states that an ideal cotorsion pair in $\mathcal{A}/\omega$ generated by a set of morphisms is complete. As another consequence, we provide some partial answers to the question about the completeness of cotorsion pairs posed by Fu, Guil Asensio, Herzog and Torrecillas.