Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T12:39:38.012Z Has data issue: false hasContentIssue false

Gross-Hopkins duality and the Gorenstein condition

Published online by Cambridge University Press:  24 September 2010

W. G. Dwyer
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, [email protected]
J. P. C. Greenlees
Affiliation:
Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH, [email protected]
S. B. Iyengar
Affiliation:
Department of Mathematics, University of Nebraska, Lincoln, NE 68588, [email protected]
Get access

Abstract

Gross and Hopkins have proved that in chromatic stable homotopy, Spanier-Whitehead duality nearly coincides with Brown-Comenetz duality. We give a conceptual interpretation of this phenomenon in terms of a Gorenstein condition [8] for maps of ring spectra.

Type
Research Article
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Angeltveit, V., Uniqueness of Morava K-theory, Compos. Math. (2010), to appear.Google Scholar
2.Atiyah, M. F., Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291310.Google Scholar
3.Avramov, L. L., Iyengar, S. B., and Lipman, J., Reflexivity and rigidity for complexes. I. Commutative rings, Algebra Number Theory 4 (2010), no. 1, 4786.CrossRefGoogle Scholar
4.Avramov, L. L., Iyengar, S. B., Lipman, J., and Nayak, S., Reduction of derived Hochschild functors over commutative algebras and schemes, Adv. Math. 223 (2010), no. 2, 735772.CrossRefGoogle Scholar
5.Blanc, D., Dwyer, W. G., and Goerss, P. G., The realization space of a Π-algebra: a moduli problem in algebraic topology, Topology 43 (2004), no. 4, 857892.CrossRefGoogle Scholar
6.Dwyer, W., Greenlees, J. P. C., and Iyengar, S., Finiteness in derived categories of local rings, Comment. Math. Helv. 81 (2006), no. 2, 383432.CrossRefGoogle Scholar
7.Dwyer, W. G. and Greenlees, J. P. C., Complete modules and torsion modules, Amer. J. Math. 124 (2002), no. 1, 199220.CrossRefGoogle Scholar
8.Dwyer, W. G., Greenlees, J. P. C., and Iyengar, S. B., Duality in algebra and topology, Adv. Math. 200 (2006), no. 2, 357402.Google Scholar
9.Elmendorf, A. D., Kriz, I., Mandell, M. A., and May, J. P., Rings, modules, and algebras in stable homotopy theory, American Mathematical Society, Providence, RI, 1997, With an appendix by M. Cole.Google Scholar
10.Goerss, P. G. and Hopkins, M. J., Moduli spaces of commutative ring spectra, Structured ring spectra, London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151200.Google Scholar
11.Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI, 2003.Google Scholar
12.Hopkins, M. J. and Gross, B. H., Equivariant vector bundles on the Lubin-Tate moduli space, Topology and representation theory (Evanston, IL, 1992), Contemp. Math. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 2388.Google Scholar
13.Hopkins, M. J. and Gross, B. H., The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 7686.Google Scholar
14.Hopkins, M. J., Mahowald, M., and Sadofsky, H., Constructions of elements in Picard groups, Topology and representation theory (Evanston, IL, 1992), Contemp. Math. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 89126.Google Scholar
15.Hovey, M., Shipley, B., and Smith, J., Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149208.CrossRefGoogle Scholar
16.Hovey, M. and Strickland, N. P., Morava K-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii + 100.Google Scholar
17.Huang, I.-C., Pseudofunctors on modules with zero-dimensional support, Mem. Amer. Math. Soc. 114 (1995), no. 548, xii+53.Google Scholar
18.Klein, J. R., The dualizing spectrum of a topological group, Math. Ann. 319 (2001), no. 3, 421456.Google Scholar
19.Matsumura, H., Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid.Google Scholar
20.Strickland, N. P., Gross-Hopkins duality, Topology 39 (2000), no. 5, 10211033.CrossRefGoogle Scholar