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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: E. Hosseini and Sh. Salarian
Journal: Proc. Amer. Math. Soc. 141 (2013), 753-762
MSC (2010): Primary 18E30, 16E40, 16E05, 13D05, 14F05
Published electronically: August 7, 2012
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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}$ $ ({\rm dg}$- $ {\rm Cof}X))$ is a complete cotorsion theory in $ \mathbf {K}(\operatorname {Flat} X)$, where $ \mathbf {K}$ $ ({\rm dg}$- $ {\rm 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.


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Additional Information

E. Hosseini
Affiliation: Department of Mathematics, University of Isfahan, Isfahan, Iran
Email: e.hosseini@sci.ui.ac.ir

Sh. Salarian
Affiliation: Department of Mathematics, University of Isfahan, P.O. Box 81746-73441, Isfahan, Iran – and – School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O. Box 19395-5746, Tehran, Iran
Email: salarian@ipm.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11364-4
PII: S 0002-9939(2012)11364-4
Keywords: Cotorsion theory, quasi-coherent sheaves, homotopy category, adjoint functors, dualizing complex
Received by editor(s): May 8, 2011
Received by editor(s) in revised form: July 11, 2011, and July 12, 2011
Published electronically: August 7, 2012
Additional Notes: This research was in part supported by a grant from IPM, No. 90130218
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society