The Homotopy Category of Cotorsion Flat Modules

Abstract

This paper aims at studying the homotopy category of cotorsion flat left modules ${{\mathbb {K}}({\mathrm{CotF}}\text {-}R)}$ over a ring R. We prove that if R is right coherent, then the homotopy category ${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$ of dg-cotorsion complexes of flat R-modules is compactly generated. This uses firstly the existence of cotorsion flat preenvelopes over such rings and, secondly, the existence of a complete cotorsion pair $({{\mathbb {K}}_{\mathrm{p}}({\mathrm{Flat}}\text {-}R)}, {\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R))$ in the homotopy category ${{\mathbb {K}}({\mathrm{Flat}}\text {-}R)}$ of complexes of flat R-modules, for arbitrary R. In the setting of quasi coherent sheaves over a Noetherian scheme, this cotorsion pair was discovered in the literature. However, we use a more elementary argument that gives this cotorsion pair for arbitrary R. Next we deal with cotorsion flat resolutions of complexes and define and study the notion of cotorsion flat dimension for complexes of flat R-modules. We also obtain an equivalence ${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\approx {{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$ of triangulated categories where ${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$ is the homotopy category of projective R-modules. Combined with the aforementioned result, this recovers a result from Neeman, asserting the compact generation of ${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$ over right coherent R. Also we get the unbounded derived category ${\mathbb {D}} (R)$ of R as a Verdier quotient of ${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$.