The flat model structure on $\mathbf {Ch}(R)$
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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 $N$.References
- S. Tempest Aldrich, Edgar E. Enochs, J. R. García Rozas, and Luis Oyonarte, Covers and envelopes in Grothendieck categories: flat covers of complexes with applications, J. Algebra 243 (2001), no. 2, 615–630. MR 1850650, DOI 10.1006/jabr.2001.8821
- W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126. MR 1361887, DOI 10.1016/B978-044481779-2/50003-1
- L. Bican, R. El Bashir, and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385–390. MR 1832549, DOI 10.1017/S0024609301008104
- Paul C. Eklof and Jan Trlifaj, How to make Ext vanish, Bull. London Math. Soc. 33 (2001), no. 1, 41–51. MR 1798574, DOI 10.1112/blms/33.1.41
- E. Enochs, S. Estrada, J.R. García-Rozas, and L. Oyonarte, Flat covers of quasi-coherent sheaves, preprint, 2000.
- Edgar E. Enochs and Overtoun M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. MR 1753146, DOI 10.1515/9783110803662
- Edgar Enochs and Luis Oyonarte, Flat covers and cotorsion envelopes of sheaves, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1285–1292. MR 1879949, DOI 10.1090/S0002-9939-01-06190-1
- Edgar E. Enochs and J. R. García Rozas, Tensor products of complexes, Math. J. Okayama Univ. 39 (1997), 17–39 (1999). MR 1680739
- J. R. García Rozas, Covers and envelopes in the category of complexes of modules, Chapman & Hall/CRC Research Notes in Mathematics, vol. 407, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1693036
- Pierre Antoine Grillet, Algebra, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1689024
- Mark Hovey, Cotorsion theories, model category structures, and representation theory, preprint, 2000.
- Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
- A. Joyal, Letter to A. Grothendieck, 1984.
- Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432, DOI 10.1007/BFb0097438
- Luigi Salce, Cotorsion theories for abelian groups, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977) Academic Press, London-New York, 1979, pp. 11–32. MR 565595
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- Robert Wisbauer, Foundations of module and ring theory, Revised and translated from the 1988 German edition, Algebra, Logic and Applications, vol. 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991. A handbook for study and research. MR 1144522
- Jinzhong Xu, Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. MR 1438789, DOI 10.1007/BFb0094173
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): 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
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 3369-3390
- MSC (2000): Primary 55U35, 18G35, 18G15
- DOI: https://doi.org/10.1090/S0002-9947-04-03416-6
- MathSciNet review: 2052954