The flat model structure on complexes of sheaves
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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 $\mathbf {Sh}(\mathcal {O})$.References
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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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2216249