nLab Landweber-Novikov theorem

Contents

Context

Higher algebra

higher algebra

universal algebra

Contents

Idea

The Landweber-Novikov theorem describes the structure of Hopf algebroid over a commutative base on the dual generalized Steenrod algebra of the complex cobordism cohomology theory spectrum MU.

By Quillen's theorem on MU we have $\pi_\bullet(MU) \simeq L$, the Lazard ring. Moreover the dual Steenrod algebra is

$MU_{\bullet}(MU) \simeq L B \coloneqq L \otimes \mathbb{Z}[b_1, b_2, \cdots]$

where on the right we have the polynomial ring on generators $b_i$ in degree $2i$.

Here $(L, L B)$ carries a natural structure of a Hopf algebroid over a commutative base (and apparently this example is what made Haynes Miller introduce Hopf algebroids in the first place). The Landweber-Novikov theorem asserts that this is the Hopf algebroid structure on the dual MU-Steenrod algebra

$(\pi_\bullet(MU), MU_\bullet(MU)) \simeq (L, L B) \,.$

The analog for BP is the Adams-Quillen theorem.

References

The original articles are

The Hopf algebroid structure on $(L, L B)$ is reviewed in

The Landweber-Novikov theorem is reviewed in

Last revised on February 5, 2016 at 05:36:02. See the history of this page for a list of all contributions to it.