The Homotopy Category of Cotorsion Flat Modules
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The Homotopy Category of Cotorsion Flat Modules Hossein Eshraghi1 · Ali Hajizamani2 Received: 13 October 2019 / Accepted: 6 October 2020 © Springer Nature B.V. 2020
Abstract This paper aims at studying the homotopy category of cotorsion flat left modules K(CotF-R) over a ring R. We prove that if R is right coherent, then the homotopy category K(dg-CotF-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 (Kp (Flat-R), K(dg-CotF-R)) in the homotopy category K(Flat-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 K(dg-CotF-R) ≈ K(Proj-R) of triangulated categories where K(Proj-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 K(Proj-R) over right coherent R. Also we get the unbounded derived category D(R) of R as a Verdier quotient of K(dg-CotF-R). Keywords Homotopy category · Compactly generated triangulated category · Cotorsion pair · Cover and envelope · Cotorsion flat resolution Mathematics Subject Classification 18E30 · 55U35 · 18G35 · 18G20 · 16E05
Communicated by Henning Krause.
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Hossein Eshraghi [email protected] Ali Hajizamani [email protected]
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Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, PO Box 87317-51167, Kashan, Iran
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Department of Mathematics, University of Hormozgan, P.O. Box: 3995, Bandar Abbas, Iran
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H. Eshraghi, A. Hajizamani
1 Introduction The theory of compactly generated triangulated categories, and in particular homotopy categories, has its origins in topology. As a powerful language to simplify the statement and the proofs of some classical results [21,22], its study has become interesting during last two decades. In this context, we may mention the papers [15,17,18,24,25] in which the problem of compact generation of K(Proj-R) and K(Inj-R), the homotopy categories formed respectively by projective and injective modules over a ring R, has been discussed; see also [12]. In particular, in [24], Neeman proves that K(Proj-R) is compactly generated whenever R is right coherent. Certainly, one of the key features of compactly generated triangulated categories is that they fulfil Brown Representability Theorem [23]. This is a nice tool to prove the existence of certain adjoints to the functors between triangulated categories. An example of this situation might be found in [16] where the existence of a particular adjoint functor leads to the existence of Gorenstein projecti
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