Ideal approximation theory in Frobenius categories
Dandan Sun, Zhongsheng Tan, Qikai Wang, Haiyan Zhu
TL;DR
This work develops ideal approximation theory in Frobenius categories by relating special precovering/preenveloping ideals in the category $\mathcal{A}$ to their counterparts in the stable category $\mathcal{A}/\omega$. Leveraging a Breaz-Modoi-type result, it establishes that certain ideals with $1_{X}\in\mathcal{I}$ for all projectives $X$ are automatically special, and shows that completeness of ideal cotorsion pairs in $\mathcal{A}$ is equivalent to precovering/preenveloping properties, with transfer to $\mathcal{A}/\omega$. This yields an ideal version of the Bongartz-Eklof-Trlifaj lemma in the stable category for sets of morphisms and yields partial answers to longstanding questions on the completeness of cotorsion pairs in this framework. The results extend classical cotorsion theory to the realm of ideals, providing a robust toolset for approximation theory in Frobenius and related triangulated settings, with potential implications for derived/homotopy-theoretic contexts.
Abstract
Let $\mathcal{A}$ be a Frobenius category and $ω$ 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 $\mathcal{A}/ω$ 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 $X\inω$ 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 $\mathcal{A}/ω$, which states that an ideal cotorsion pair in $\mathcal{A}/ω$ 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.