Abstract
For a tensor triangulated category which is well generated in the sense of Neeman, it is shown that the collection of Bousfield classes forms a set. This set has a natural structure of a complete lattice which is then studied, using the notions of stratification and support.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alonso Tarrí o L., Jeremías López A., Souto Salorio M.J.: Bousfield localization on formal schemes. J. Algebra 278(2), 585–610 (2004)
Balmer P.: The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. 588, 149–168 (2005)
Balmer P., Favi G.: Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. 102(6), 1161–1185 (2011)
Benson D.J., Iyengar S.B., Krause H.: Local cohomology and support for triangulated categories. Ann. Sci. École Norm. Sup. 41(4), 573–619 (2008)
Benson D.J., Iyengar S.B., Krause H.: Stratifying modular representations of finite groups. Ann. Math. (1) 174, 1643–1684 (2011)
Benson D.J., Iyengar S.B., Krause H.: Stratifying triangulated categories. J. Topol. 4, 641–666 (2011)
Benson, D.J., Iyengar, S.B., Krause, H.: Colocalizing subcategories and cosupport. J. Reine Angew. Math. doi:10.1515/crelle.2011.180. arXiv:1008.3701
Borceux, F.: Handbook of categorical algebra. 3. In: Encyclopedia of Mathematics and its Applications, vol. 52. Cambridge University Press, Cambridge (1994)
Bousfield, A.K.: The Boolean algebra of spectra. Comm. Math. Helv. 54, 368–377 (1979) [Correction: Comm. Math. Helv. 58, 599–600 (1983)]
Buan A.B., Krause H., Solberg Ø.: Support varieties: an ideal approach. Homol. Homot. Appl. 9(1), 45–74 (2007)
Casacuberta, C., Gutiérrez, J.J., Rosický, J.: A generalization of Ohkawa’s theorem. (preprint 2012). arXiv:1203.6395
Christensen J.D., Strickland N.P.: Phantom maps and homology theories. Topology 37(2), 339–364 (1998)
Dwyer W.G., Palmieri J.H.: Ohkawa’s theorem: there is a set of Bousfield classes. Proc. Am. Math. Soc. 129(3), 881–886 (2001)
Dwyer W.G., Palmieri J.H.: The Bousfield lattice for truncated polynomial algebras. Homol. Homot. Appl. 10(1), 413–436 (2008)
Gabriel P.: Des catégories abéliennes. Bull. Soc. Math. Fr. 90, 323–448 (1962)
Hochster M.: Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 142, 43–60 (1969)
Hovey, M., Palmieri, J.H.: The structure of the Bousfield lattice. In: Homotopy Invariant Algebraic Structures (Baltimore, MD, 1998). Contemp. Math., vol. 239, pp. 175–196. Amer. Math. Soc., Providence, RI (1999)
Hovey, M., Palmieri, J.H., Strickland, N.P.: Axiomatic stable homotopy theory. In: Mem. Amer. Math. Soc., vol. 610. Amer. Math. Soc., Providence (1997)
Iyengar, S.B.: The Bousfield lattice of the stable module category of a finite group. In: Representation Theory of Quivers and Finite Dimensional Algebras (Oberwolfach, 2011), Oberwolfach Reports, vol. 8, No. 1. European Mathematical Society, Zürich (2011)
Johnstone, P.T.: Stone spaces, reprint of the 1982 edition. In: Cambridge Studies in Advanced Mathematics, Vol. 3, Cambridge University Press, Cambridge (1986)
Kock, J.: Spectra, supports, and Hochster duality. Letter to G. Favi and P. Balmer, December 2007. http://mat.uab.es/~kock/cat/spec.pdf
Krause H.: Smashing subcategories and the telescope conjecture—an algebraic approach. Invent. Math. 139(1), 99–133 (2000)
Krause H.: On Neeman’s well generated triangulated categories. Doc. Math. 6, 121–126 (2001)
Krause, H.: Localization theory for triangulated categories. In: Triangulated Categories. London Math. Soc. Lecture Note Ser., vol. 375, pp. 161–235, Cambridge University Press, Cambridge (2010)
Lewis, L.G., Jr. et al.: Equivariant stable homotopy theory. In: Lecture Notes in Mathematics, vol. 1213. Springer, Berlin (1986)
Neeman A.: The chromatic tower for D(R). Topology 31(3), 519–532 (1992)
Neeman A.: The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4) 25(5), 547–566 (1992)
Neeman A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9(1), 205–236 (1996)
Neeman, A.: Triangulated categories. In: Annals of Math. Studies, vol. 148. Princeton University Press, Princeton (2001)
Ohkawa T.: The injective hull of homotopy types with respect to generalized homology functors. Hiroshima Math. J. 19(3), 631–639 (1989)
Ravenel, D.C.: Nilpotence and periodicity in stable homotopy theory. In: Annals of Mathematics Studies, vol. 128. Princeton University Press, Princeton, NJ (1992)
Rota G.-C.: The many lives of lattice theory. Not. Am. Math. Soc. 44(11), 1440–1445 (1997)
Stevenson, G.: Support theory via actions of tensor triangulated categories. J. Reine Angew. Math. doi:10.1515/crelle-2012-0025. arXiv:1105.4692
Stevenson, G.: Subcategories of singularity categories via tensor actions (preprint 2011). arXiv:1105:4698
Stone M.H.: Topological representations of distributive lattices and Brouwerian logics. Časopis pro Pěstování Matematiky a Fysiky 67, 1–25 (1937)
Strickland, N.P.: Axiomatic stable homotopy. In: Axiomatic, Enriched and Motivic Homotopy Theory. NATO Sci. Ser. II Math. Phys. Chem., vol. 131, pp. 69–98, Kluwer Acad. Publ., Dordrecht (2004)
Thomason R.W.: The classification of triangulated subcategories. Compos. Math. 105(1), 1–27 (1997)
Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque, vol. 239. Société Math. de France (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of SI was partly supported by National Science Foundation Grant DMS 0903493.
Rights and permissions
About this article
Cite this article
Iyengar, S.B., Krause, H. The Bousfield lattice of a triangulated category and stratification. Math. Z. 273, 1215–1241 (2013). https://doi.org/10.1007/s00209-012-1051-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-012-1051-7