A cotorsion theory in the homotopy category of flat quasi-coherent sheaves
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- by E. Hosseini and Sh. Salarian PDF
- Proc. Amer. Math. Soc. 141 (2013), 753-762 Request permission
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 $(\mathbf {K}_{\operatorname {p}}({\operatorname {Flat}} X)$, $\mathbf {K}$ $(\textrm {dg}$-$\textrm {Cof}X))$ is a complete cotorsion theory in $\mathbf {K}(\operatorname {Flat} X)$, where $\mathbf {K}$ $(\textrm {dg}$-$\textrm {Cof}X)$ 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.References
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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
- 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
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 753-762
- MSC (2010): Primary 18E30, 16E40, 16E05, 13D05, 14F05
- DOI: https://doi.org/10.1090/S0002-9939-2012-11364-4
- MathSciNet review: 3003669