Recollements from Cotorsion Pairs

Abstract

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 $\mathrm{Ch}(\mathcal{B})$, 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 $\mathrm{Ch}(\mathcal{G})$ with terms in $\mathcal{B}$. In this case the derived category $\mathcal{D}(\mathcal{B})$ fits into a recollement $\frac{ex\mathcal{B}}{\sim }\begin{array}{c}\leftarrow \to \leftarrow \end{array}K(\mathcal{B})\begin{array}{c}\leftarrow \to \leftarrow \end{array}\frac{\mathrm{Ch}(\mathcal{B})}{ex\mathcal{B}}$. We will explore the conditions under which $\mathrm{ex}\mathcal{B}=\tilde{\mathcal{B}}$ and provide many examples. Symmetrically, we prove analogous results for the exact category $\mathcal{A}$.

Added Lemma 1.2 and fixed statement of Proposition 2.2

References in corpus (3)

• Derived equivalences induced by big cotilting modules
• On purity and applications to coderived and singularity categories
• Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves