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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The flat model structure on $\mathbf{Ch}(R)$


Author: James Gillespie
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3369-3390
MSC (2000): Primary 55U35, 18G35, 18G15
Published electronically: January 29, 2004
MathSciNet review: 2052954
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Abstract: Given a cotorsion pair $(\mathcal{A},\mathcal{B})$ in an abelian category $\mathcal{C}$ with enough $\mathcal{A}$ objects and enough $\mathcal{B}$ objects, we define two cotorsion pairs in the category $\mathbf{Ch(\mathcal{C})}$ of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when $(\mathcal{A},\mathcal{B})$ is hereditary. We then show that both of these induced cotorsion pairs are complete when $(\mathcal{A},\mathcal{B})$ is the ``flat'' cotorsion pair of $R$-modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new ``flat'' model category structure on $\mathbf{Ch}(R)$. In the last section we use the theory of model categories to show that we can define $\operatorname{Ext}^n_R(M,N)$using a flat resolution of $M$ and a cotorsion coresolution of $N$.


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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: http://dx.doi.org/10.1090/S0002-9947-04-03416-6
PII: S 0002-9947(04)03416-6
Received by editor(s): October 1, 2002
Received by editor(s) in revised form: May 13, 2003
Published electronically: January 29, 2004
Additional Notes: The author thanks Mark Hovey of Wesleyan University
Article copyright: © Copyright 2004 American Mathematical Society