Co-t-structures, cotilting and cotorsion pairs

Abstract

AbstractLet $\textsf{T}$ be a triangulated category with shift functor $\Sigma \colon \textsf{T} \to \textsf{T}$ . Suppose $(\textsf{A},\textsf{B})$ is a co-t-structure with coheart $\textsf{S} = \Sigma \textsf{A} \cap \textsf{B}$ and extended coheart $\textsf{C} = \Sigma^2 \textsf{A} \cap \textsf{B} = \textsf{S}* \Sigma \textsf{S}$ , which is an extriangulated category. We show that there is a bijection between co-t-structures $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ in $\textsf{T}$ such that $\textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}$ and complete cotorsion pairs in the extended coheart $\textsf{C}$ . In the case that $\textsf{T}$ is Hom-finite, $\textbf{k}$ -linear and Krull–Schmidt, we show further that there is a bijection between complete cotorsion pairs in $\textsf{C}$ and functorially finite torsion classes in $\textsf{mod}\, \textsf{S}$ .