The flat model structure on complexes of sheaves

Gillespie, James

doi:10.1090/S0002-9947-06-04157-2

The flat model structure on complexes of sheaves
HTML articles powered by AMS MathViewer

by James Gillespie PDF

Trans. Amer. Math. Soc. 358 (2006), 2855-2874 Request permission

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

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
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
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
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
Edgar Enochs, S. Estrada, J. R. García Rozas, and L. Oyonarte, Flat covers in the category of quasi-coherent sheaves over the projective line, Comm. Algebra 32 (2004), no. 4, 1497–1508. MR 2100370, DOI 10.1081/AGB-120028794
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
James Gillespie, The flat model structure on $\textrm {Ch}(R)$, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3369–3390. MR 2052954, DOI 10.1090/S0002-9947-04-03416-6
Pierre Antoine Grillet, Algebra, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1689024
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093, DOI 10.1007/BFb0080482
Mark Hovey, Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2441–2457. MR 1814077, DOI 10.1090/S0002-9947-01-02721-0
Mark Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), no. 3, 553–592. MR 1938704, DOI 10.1007/s00209-002-0431-9
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.
T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
Shigeru Iitaka, Algebraic geometry, North-Holland Mathematical Library, vol. 24, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties. MR 637060, DOI 10.1007/978-1-4613-8119-8
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
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
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953, DOI 10.1007/978-3-642-66066-5
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