Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Model category structures on chain complexes of sheaves


Author: Mark Hovey
Journal: Trans. Amer. Math. Soc. 353 (2001), 2441-2457
MSC (2000): Primary 18F20, 14F05, 18E15, 18E30, 18G35, 55U35
Published electronically: January 3, 2001
MathSciNet review: 1814077
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

The unbounded derived category of a Grothendieck abelian category is the homotopy category of a Quillen model structure on the category of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently by Beke. However, in most cases of interest, such as the category of sheaves on a ringed space or the category of quasi-coherent sheaves on a nice enough scheme, the abelian category in question also has a tensor product. The injective model structure is not well-suited to the tensor product. In this paper, we consider another method for constructing a model structure. We apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.


References [Enhancements On Off] (What's this?)

  • [Bek99] Tibor Beke, Sheafifiable homotopy model categories, preprint, 1999.
  • [Chr98] J. Daniel Christensen, Derived categories and projective classes, preprint, 1998.
  • [DS95] W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126. MR 1361887, 10.1016/B978-044481779-2/50003-1
  • [Gro57] Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119–221 (French). MR 0102537
  • [Gro60] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228. MR 0217083
  • [Har66] Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. MR 0222093
  • [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [Hov98] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
  • [HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 1388895, 10.1090/memo/0610
  • [Joy84] A. Joyal, Letter to A. Grothendieck, 1984.
  • [KS90] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006
  • [Lip98] Joseph Lipman, Notes on derived categories and derived functors, preprint, 1998.
  • [MP92] I. Moerdijk and D. A. Pronk, Cohomology of sheaves, lecture notes, 1992.
  • [Qui67] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
  • [SGA71] Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
  • [Spa88] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640
  • [SS97] Stefan Schwede and Brooke Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), 491-511. CMP 2000:07
  • [Ste75] Bo Stenström, Rings of quotients, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217; An introduction to methods of ring theory. MR 0389953
  • [TLS99] L. Alonso Tarrío, A. Jeremias López, and M. J. Souto Salorio, Localizations in categories of complexes and unbounded resolutions, Canad. J. Math. 52 (2000), 225-247. CMP 2000:12

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 18F20, 14F05, 18E15, 18E30, 18G35, 55U35

Retrieve articles in all journals with MSC (2000): 18F20, 14F05, 18E15, 18E30, 18G35, 55U35


Additional Information

Mark Hovey
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Email: hovey@member.ams.org

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02721-0
Received by editor(s): February 24, 2000
Published electronically: January 3, 2001
Article copyright: © Copyright 2001 American Mathematical Society