Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-08-31T03:57:24.269Z Has data issue: false hasContentIssue false
  • English
  • Français

Projective and Injective Hopf Algebras Over the Dyer-Lashof Algebra

Published online by Cambridge University Press:  20 November 2018

Paul G. Goerss
Paul G. Goerss*
Affiliation:
Department of Mathematics University of Washington Seattle, Washington 98195 U.S.A
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to discuss the existence, structure, and properties of certain projective and injective Hopf algebras in the category of Hopf algebras that support the structure one expects on the homology of an infinite loop space. As an auxiliary project, we show that these projective and injective Hopf algebras can be realized as the homology of infinite loop spaces associated to spectra obtained from Brown-Gitler spectra by Spanier-Whitehead duality and Brown-Comenetz duality, respectively. We concentrate mainly on indecomposable projectives and injectives, and we work only at the prime 2.

Keywords

55P3555N99Hopf algebrasDyer-Lashof algebraprojective and injective objectsBrown-Gitler spectra

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 5 , 01 October 1993 , pp. 944 - 976
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Brown, E.H., Jr. and Comenetz, M., Pontrjagin duality for generalized homology and cohomology theories, Amer J. Math. 98(1976), 1–27.Google Scholar
2. Brown, E.H., Jr. and Gitler, S.,A spectrum whose cohomology is a certain cyclic module over the Steenrod Algebra, Topology 12(1973), 283–295.Google Scholar
3. Cohen, F.R., Lada, T.J., and May, J.P., The Homology of Iterated Loop Spaces, Lecture Notes in Mathematics 533, Springer, Berlin, 1976.Google Scholar
4. Goerss, P.G., Unstable projectives and stable Ext, Proc. London Math. Soc. 53(1986), 539–561.Google Scholar
5. Goerss, P., Lannes, J., and Morel, F., Vecteurs de Witt non-commutatifs et representabilité de Vhomologie modulop, Invent. Math. 108(1992), 163–227.Google Scholar
6. Lannes, J., Sur le n-dual du n-ème spectre de Brown-Gitler, Math Z. 199(1988), 29–42.Google Scholar
7. Lannes, J. et Zarati, S., Sur les ᥔ -injectifs, Ann. Sci. Ecole Normale Sup. 19(1986), 1–31.Google Scholar
8. MacLane, S., Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer,Berlin, 1971.Google Scholar
9. Madsen, I., On the action of the Dyer-Lashof algebra in H*G, Pacific J. Math. 60(1975), 235–275.Google Scholar
10. Miller, H., A spectral sequence for the homology of an infinite delooping, Pacific J. Math. 79(1978), 139—155.Google Scholar
11. Miller, H. , The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120(1984), 39–87.Google Scholar
12. Schoeller, C., Etude de la catégorie des algebras de Hopf commutative connexe sur un corps, Manusc. Math. 3(1970), 133–155.Google Scholar
13. Steiner, R., Decompositions of groups of units in ordinary cohomology, Quart. J. Math. 30(1979), 483–494.Google Scholar
14. Sweedler, M., Hopf Algebras, Benjamin, W.A., New York, 1969.Google Scholar
15. Wilson, W.S., Brown-Peterson Homology: An Introduction and Sampler, Regional Conf. Series in Math 48, A.M.S., Providence, 1982.Google Scholar