A cotorsion theory in the homotopy category of flat quasi-coherent sheaves

Abstract:

Let $X$ be a Noetherian scheme, $\mathbf {K}(\operatorname {Flat} X)$ be the homotopy category of flat quasi-coherent $\mathcal {O}_X$-modules and $\mathbf {K}_{\operatorname {p}}({\operatorname {Flat}} X)$ be the homotopy category of all flat complexes. It is shown that the pair $(\mathbf {K}_{\operatorname {p}}({\operatorname {Flat}} X)$, $\mathbf {K}$ $(\textrm {dg}$-$\textrm {Cof}X))$ is a complete cotorsion theory in $\mathbf {K}(\operatorname {Flat} X)$, where $\mathbf {K}$ $(\textrm {dg}$-$\textrm {Cof}X)$ is the essential image of the homotopy category of dg-cotorsion complexes of flat modules. Then we study the homotopy category $\mathbf {K}$($\operatorname {dg}$-$\operatorname {Cof}X$). We show that in the affine case, this homotopy category is equal with the essential image of the embedding functor $j_* : \mathbf {K}({\operatorname {Proj}}R) \longrightarrow \mathbf {K}({\operatorname {Flat}}R)$ which has been studied by Neeman in his recent papers. Moreover, we present a condition for the inclusion $\mathbf {K}$($\operatorname {dg}$-$\operatorname {Cof}X$) $\subseteq \mathbf {K}(\operatorname {{Cof}} X)$ to be an equality, where $\mathbf {K}(\operatorname {{Cof}} X)$ is the essential image of the homotopy category of complexes of cotorsion flat sheaves.