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

Abstract

Let X be a Noetherian scheme, K(FlatX) be the homotopy category of flat quasi-coherent OX-modules and Kp(FlatX) be the homotopy category of all flat complexes. It is shown that the pair (Kp(FlatX), K (dg- CofX)) is a complete cotorsion theory in K(FlatX), where K (dg-CofX) is the essential image of the homotopy category of dg-cotorsion complexes of flat modules. Then we study the homotopy category K(dg-Cof X). We show that in the affine case, this homotopy category is equal with the essential image of the embedding functor j*: K(ProjR) → K(FlatR) which has been studied by Neeman in his recent papers. Moreover, we present a condition for the inclusion K(dg-Cof X) ⊆ K(Cof X) to be an equality, where K(Cof X) is the essential image of the homotopy category of complexes of cotorsion flat sheaves. © 2012, American Mathematical Society.