Construction of ideal cotorsion pairs via recollements of triangulated categories

TL;DR

This paper explores how complete ideal cotorsion pairs in triangulated categories can be constructed and transferred through recollements, providing a systematic method to generate new cotorsion pairs.

Contribution

It introduces a method to construct and transfer complete ideal cotorsion pairs across recollements of triangulated categories, expanding the toolkit for their analysis.

Findings

Complete ideal cotorsion pairs in a category induce pairs in subcategories.

Pairs in subcategories combine to form new pairs in the larger category.

Method enables systematic construction of cotorsion pairs in complex categories.

Abstract

Let $(\mathcal{T}',\mathcal{T},\mathcal{T}'')$ be a recollement of triangulated categories.A complete ideal cotorsion pair in $\mathcal{T}$ induces complete ideal cotorsion pairs in $\mathcal{T}'$ and $\mathcal{T}''$. In addition, if $(\mathcal{I}, \mathcal{I}^\perp )$ and $(\mathcal{J},\mathcal{J}^\perp)$ are two complete ideal cotorsion pairs in a triangulated category, then $(\mathcal{I}\cap\mathcal{J}, \langle\mathcal{I}^\perp,\mathcal{J}^\perp\rangle)$ is also a complete ideal cotorsion pair. By this method, starting from two complete ideal cotorsion pairs in $\mathcal{T}'$ and $\mathcal{T}''$, one can induce a family of complete ideal cotorsion pairs in $\mathcal{T}$.