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Transactions of the American Mathematical Society

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The flat model structure on complexes of sheaves


Author: James Gillespie
Journal: Trans. Amer. Math. Soc. 358 (2006), 2855-2874
MSC (2000): Primary 55U35, 18G15
DOI: https://doi.org/10.1090/S0002-9947-06-04157-2
Published electronically: February 14, 2006
MathSciNet review: 2216249
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Abstract: Let $ \mathbf{Ch}(\mathcal{O})$ be the category of chain complexes of $ \mathcal{O}$-modules on a topological space $ T$ (where $ \mathcal{O}$ is a sheaf of rings on $ T$). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on $ \mathbf{Ch}(\mathcal{O})$. As a corollary, we have a general framework for doing homological algebra in the category $ \mathbf{Sh}(\mathcal{O})$ of $ \mathcal{O}$-modules. I.e., we have a natural way to define the functors $ \operatorname{Ext}$ and $ \operatorname{Tor}$ in $ \mathbf{Sh}(\mathcal{O})$.


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Additional Information

James Gillespie
Affiliation: Department of Mathematics, 4000 University Drive, Penn State McKeesport, McKeesport, Pennsylvania 15132-7698
Email: jrg21@psu.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04157-2
Received by editor(s): January 8, 2004
Published electronically: February 14, 2006
Additional Notes: The author thanks Mark Hovey of Wesleyan University and Edgar Enochs of the University of Kentucky
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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