Recollements from Cotorsion Pairs

Given a complete hereditary cotorsion pair $(\mathcal{A},\mathcal{B})$ in a Grothendieck category $\mathcal{G}$, the derived category $\mathcal{D}(\mathcal{B})$ of the exact category $\mathcal{B}$ is defined as the quotient of the category $\mathrm{C}\mathrm{h}(\mathcal{B})$, of unbounded complexes with terms in $\mathcal{B}$, modulo the subcategory $\tilde{\mathcal{B}}$ consisting of the acyclic complexes with terms in $\mathcal{B}$ and cycles in $\mathcal{B}$. We restrict our attention to the cotorsion pairs such that $\tilde{\mathcal{B}}$ coincides with the class $ex\mathcal{B}$ of the acyclic complexes of $\mathrm{C}\mathrm{h}(\mathcal{G})$ with terms in $\mathcal{B}$. In this case the derived category $\mathcal{D}(\mathcal{B})$ fits into a recollement $\dfrac{ex$\mathcal{B}$}{\sim} \mathrel{\substack{\textstyle\leftarrow\textstyle\rightarrow\textstyle\leftarrow}} {K($\mathcal{B}$)} \mathrel{\substack{\textstyle\leftarrow\textstyle\rightarrow\textstyle\leftarrow}} {\dfrac{$\mathrm{C}\mathrm{h}$($\mathcal{B}$)}{ex$\mathcal{B}$ }}$. We will explore the conditions under which $\mathrm{e}\mathrm{x}\mathcal{B}=\tilde{\mathcal{B}}$ and provide many examples. Symmetrically, we prove analogous results for the exact category $\mathcal{A}$.