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 \({\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\).

本文旨在研究环R上的扭曲左平模块\（{{\ mathbb {K}}（{\ mathrm {CotF}} \ text {-} R）} \）的同伦类。我们证明，如果[R是正确的连贯，那么同伦类\（{\ mathbb {K}}（\ mathrm {DG} \ {文本- } \ mathrm {CotF} \ {文本- } R）\）的DG-紧凑地生成了平面R模块的扭曲复合体。这首先使用在这样的环上存在扭曲平坦的前包络，其次，使用完整的扭曲对\（（{{\ mathbb {K}} _ {\ mathrm {p}}（{\ mathrm {Flat}} \ text {-} R）}，{\ mathbb {K}}（\ mathrm {dg} \ text {-} \ mathrm {CotF} \ text {-} R））\）在同伦类别\（{{ \ mathbb {K}}（{\ mathrm {Flat}} \ text {-} R）} \）任意R的平面R-模的复数的个数。在Noetherian方案上的准相干滑轮的环境中，这种扭曲对在文献中被发现。但是，我们使用了一个更基本的参数，该参数为任意R给出了这个扭曲对。接下来，我们处理复合物的挠曲平面分辨率，并定义和研究平面R模块的复合物的挠曲平面尺寸的概念。我们还获得了等价\（{\ mathbb {K}}（\ mathrm {dg} \ text {-} \ mathrm {CotF} \ text {-} R} \ approx {{\ mathbb {K}}（{\三角类别的mathrm {Proj}} \ text {-} R）} \），其中\（{{\ mathbb {K}}（{\ mathrm {Proj}} \ text {-} R）} \）是同伦射影R的类别-模块。结合上述结果，这从Neeman恢复了一个结果，断言了在右相关R上\（{{\ mathbb {K}}（{\ mathrm {Proj}} \ text {-} R）} \）的紧凑生成。同时，我们也得到了无限的派生类\（{\ mathbb {d}}（R）\）的[R作为一个维迪尔商\（{\ mathbb {K}}（\ mathrm {DG} \文本{ - } \ mathrm {CotF} \ text {-} R）\）。