Construction of ideal cotorsion pairs via recollements of triangulated categories

Abstract

Let $({\mathcal{T}}^{\prime },\mathcal{T},{\mathcal{T}}^{\prime \prime })$ be a recollement of triangulated categories.A complete ideal cotorsion pair in $\mathcal{T}$ induces complete ideal cotorsion pairs in ${\mathcal{T}}^{\prime }$ and ${\mathcal{T}}^{\prime \prime }$. 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 $(\mathcal{I}\cap \mathcal{J},⟨{\mathcal{I}}^{\perp },{\mathcal{J}}^{\perp }⟩)$ is also a complete ideal cotorsion pair. By this method, starting from two complete ideal cotorsion pairs in ${\mathcal{T}}^{\prime }$ and ${\mathcal{T}}^{\prime \prime }$, one can induce a family of complete ideal cotorsion pairs in $\mathcal{T}$.