nLab Lazard's theorem

This page is about characterizing the Lazard ring in formal group laws. For the characterization of flat modules see instead at Lazard's criterion.

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Statement

Theorem

The Lazard ring is isomorphic to a graded polynomial ring

$$L \simeq \mathbb{Z}[t_1, t_2, \cdots]$$

with the variable $t_i$ in degree $2 i$.

A proof is spelled out in (Kochman 96, theorem 4.4.9). Another proof is indicated in (Hopkins 99, section 2), worked out in (Mathew 12, Wilson 13). Also (Lurie 10, lecture 2, theorem 4) and (Lurie 10, lecture 3).

Related theorem

• Quillen's theorem on MU

• Landweber exact functor theorem

• Landweber-Novikov theorem

References

The proof is originally due to :

• Michel Lazard, Sur les groupes de Lie Formels à un Paramètre, Bull. Soc. France 83 (1955) [numdam:BSMF_1955__83__251_0]

• A. Fröhlich, Formal group, Lecture Notes in Mathematics 74, Springer (1968)

Review includes

• Frank Adams, part II.7 of Stable homotopy and generalised homology, 1974

• Stanley Kochman, section 4.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Mike Hopkins, Complex oriented cohomology theories and the language of stacks, 1999 course notes pdf

• Akhil Mathew, Lazard’s theorem II, 2012

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 2 Lazard’s theorem (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 3 Lazard’s theorem (continued) (pdf)

• Dylan Wilson, Proof of Lazard’s theorem, lecture at 2013 Pre-Talbot Seminar, March 2013 (pdf)