Journal of the Mathematical Society of Japan

Odd primary Steenrod algebra, additive formal group laws, and modular invariants

Masateru INOUE

Full-text: Open access


We give a conceptual clarification of Milnor's theorem, which tells us the Hopf algebra structure of the stable co-operations H * H in the odd primary ordinary cohomology. Directly connecting H * H with the quasi-strict automorphism group of some 1 -dimensional additive formal group law and modular invariants, we give a new proof of this theorem of Milnor.

Article information

J. Math. Soc. Japan, Volume 58, Number 2 (2006), 311-332.

First available in Project Euclid: 1 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55S10: Steenrod algebra
Secondary: 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 55P20: Eilenberg-Mac Lane spaces

Steenrod algebra formal group laws multiplicative operations reduced power operations Eilenberg-MacLane spectrum modular invariants


INOUE, Masateru. Odd primary Steenrod algebra, additive formal group laws, and modular invariants. J. Math. Soc. Japan 58 (2006), no. 2, 311--332. doi:10.2969/jmsj/1149166777.

Export citation


  • J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, Ill.-London, 1974, pp.,x+373.
  • H. Cartan, Sur les groupes d'Eilenberg-MacLane $H(\varPi,n)$. I. Méthode des constructions, Proc. Natl. Acad. Sci. USA, 40 (1954), 467–471.
  • H. Cartan, Sur les groupes d'Eilenberg-MacLane. II, Proc. Natl. Acad. Sci. USA, 40 (1954), 704–707.
  • L. E. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc., 12 (1911), 75–98.
  • M. Hazewinkel, Formal groups and applications, Pure and Appl. Math., 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978, pp.,xxii+573.
  • M. Inoue, The Steenrod algebra and the automorphism group of additive formal group law, J. Math. Kyoto Univ., 45 (2005), No.,1, 39–55.
  • J. Milnor, The Steenrod algebra and its dual, Ann. of Math., 67 (1958), 150–171.
  • H. Mùi, Modular invariant theory and the cohomology algebras of symmetric spaces, J. Fac. Sci. Univ. Tokyo, 22 (1975), 319–369.
  • H. Mùi, Cohomology operations derived from modular invariants, Math. Z., 193 (1986), 151–163.
  • N. E. Steenrod and D. B. A. Epstein, Cohomology Operations, Ann. of Math. Stud., No.,50, Princeton Univ. Press, 1962.
  • N. Sum, Steenrod operations on the modular invariants, Kodai Math. J., 17 (1994), no.,3, 585–595.