Abstract
Topological André-Quillen homology for commutative S-algebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative S-algebras and apply it to obtain results on minimal atomic p-local S-algebras which generalise those of Baker and May for p-local spectra and simply connected spaces. We exhibit some new examples of minimal atomic commutative S-algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, J.F.: Stable Homotopy and Generalised Homology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1974)
Baker, A.J., May, J.P.: Minimal atomic complexes. Topology 43(2), 645–665 (2004)
Basterra, M.: André-Quillen cohomology of commutative S-algebras. J. Pure Appl. Algebra 144(2), 111–143 (1999)
Basterra, M., Mandell, M.: Homology and cohomology of E ∞ ring spectra. Math. Z. 249(4), 903–944 (2005)
Basterra, M., McCarthy, R.: Γ-homology, topological André-Quillen homology and stabilization. Topol. Appl. 121(3), 551–566 (2002)
Basterra, M., Richter, B.: (Co-)homology theories for commutative S-algebras. In: Structured Ring Spectra. Mathematical Society Lecture Note Series, vol. 315, pp. 115–130. Cambridge University Press, Cambridge (2004)
Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, Modules, and Algebras in Stable Homotopy Theory. Mathematical Surveys and Monographs, vol. 47. Am. Math. Soc., Providence (1997). With an appendix by M. Cole
Floyd, E.E.: Stiefel-Whitney numbers of quaternionic and related manifolds. Trans. Am. Math. Soc. 155(2), 77–94 (1971)
Gilmour, H.: Nuclear and minimal atomic S-algebras. Ph.D. Thesis, University of Glasgow (2006)
Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. Am. Math. Soc., Providence (1999)
Hu, P., Kriz, I., May, J.P.: Cores of spaces, spectra and E ∞ ring spectra. Homology Homotopy Appl. 3(2), 341–54 (2001)
Kochman, S.O.: The Symplectic Cobordism Ring, I. Mem. Am. Math. Soc., vol. 228. Am. Math. Soc., Providence (1980)
Kriz, I.: Towers of E ∞ ring spectra with an application to BP. Unpublished preprint
Kuhn, N.J.: Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces. Adv. Math. 201(2), 318–378 (2006)
Mandell, M.A.: Topological André-Quillen cohomology and E ∞ André-Quillen cohomology. Adv. Math. 177(2), 227–279 (2003)
Quillen, D.: On the (co)-homology of commutative rings. Appl. Categ. Algebra Proc. Symp. Pure Math. 17, 65–87 (1970)
Rognes, J.: Galois extensions of structured ring spectra. Mem. Am. Math. Soc. 192(898), 1–97 (2008)
Stong, R.E.: Notes on Cobordism Theory. Mathematical Notes. Princeton University Press/University of Tokyo Press, Princeton (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by B. Richter.
A. Baker was partially supported by a YFF Norwegian Research Council grant while at the University of Oslo in 2007–8, a Carnegie Trust for the Universities of Scotland grant, and Intas grants 03-51-3251 and 06-1000017-8609.
H. Gilmour was supported by an EPSRC studentship.
P. Reinhard was supported by an ORS grant.
We would like to thank M. Basterra, P. Kropholler, M. Mandell, P. May, B. Richter, J. Rognes and S. Sagave for numerous helpful comments. We are also very grateful to the referee for encouraging us to rethink significantly issues of notation and structure, thus improving the structure of the paper.
Rights and permissions
About this article
Cite this article
Baker, A., Gilmour, H. & Reinhard, P. Topological André-Quillen homology for cellular commutative S-algebras. Abh. Math. Semin. Univ. Hambg. 78, 27–50 (2008). https://doi.org/10.1007/s12188-008-0005-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-008-0005-9