Co-t-structures, cotilting and cotorsion pairs

TL;DR

This paper explores the relationships between co-t-structures, cotilting objects, and cotorsion pairs within triangulated categories, establishing bijections that connect these concepts and extend classical results.

Contribution

It introduces a bijection between co-t-structures with certain inclusions and cotorsion pairs in an extended coheart, and further relates these to torsion pairs in module categories under specific conditions.

Findings

Bijection between co-t-structures and cotorsion pairs in the extended coheart.

In Hom-finite, $ extbf{k}$-linear, Krull-Schmidt categories, a bijection with torsion pairs in module categories.

Extension of classical correspondences in triangulated and extriangulated categories.

Abstract

Let $\mathsf{T}$ be a triangulated category with shift functor $\Sigma \colon \mathsf{T} \to \mathsf{T}$. Suppose $(\mathsf{A},\mathsf{B})$ is a co-t-structure with coheart $\mathsf{S} = \Sigma \mathsf{A} \cap \mathsf{B}$ and extended coheart $\mathsf{C} = \Sigma^2 \mathsf{A} \cap \mathsf{B} = \mathsf{S} * \Sigma \mathsf{S}$, which is an extriangulated category. We show that there is a bijection between co-t-structures $(\mathsf{A}',\mathsf{B}')$ in $\mathsf{T}$ such that $\mathsf{A} \subseteq \mathsf{A}' \subseteq \Sigma \mathsf{A}$ and complete cotorsion pairs in the extended coheart $\mathsf{C}$. In the case that $\mathsf{T}$ is Hom-finite, $\mathbf{k}$-linear and Krull-Schmidt, we show further that there is a bijection between complete cotorsion pairs in $\mathsf{C}$ and functorially finite torsion pairs in $\mathsf{mod}\, \mathsf{S}$.