# nLab Landweber exact functor theorem

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

Every complex oriented cohomology theory induces a formal group law from its first Conner-Floyd Chern class. Moreover, Quillen's theorem on MU together with Lazard's theorem say that the cohomology ring $\pi_\bullet(M U)$ of complex cobordism cohomology MU is the classifying ring for formal group laws.

The Landweber exact functor theorem (Landweber 76) says that, conversely, forming the tensor product of complex cobordism cohomology theory (MU) with a Landweber exact ring via some formal group law yields a cohomology theory.

By the Brown representability theorem this defines a spectrum and the spectra arising this way are called Landweber exact spectra. (e.g. Lurie lect 17, p.2).

There is an analogue of the statement also for even 2-periodic cohomology theories, with MU replaced by MP (Lurie lecture 18, prop. 11).

## Statement

By evaluation on $X = B U(1)$ the classifying space for complex line bundles (the circle 2-group), every multiplicative complex oriented cohomology theory $E$ gives rise to a formal group over the graded ring $\pi_\bullet(E)$ (see there).

Conversely, given a formal group $F$ over a graded ring $R$, one can ask whether this arises from some $E$ in this way.

Now since the Lazard ring $L$ classifies formal groups in that $F$ is equivalently given by a ring homomorphism $L \longrightarrow R$ and since moreover Quillen's theorem on MU identifies $L$ with the cohomology ring of complex oriented cohomology theory $L \simeq MU_\bullet(\ast)$, one may consider the functor

$X \mapsto E_\bullet(X) \coloneqq MU_\bullet(X) \underset{L}{\otimes} R \,.$

By construction this is such that $E_\bullet(\ast) \simeq R$. But this functor may fail to satisfy the axioms of a generalized homology theory, hence may fail to be Brown represented by an actual spectrum $E$.

Landweber exactness is a sufficient condition on $R$ for $E \coloneqq MU_\bullet(-)\otimes_L R$ to be indeed a generalized homology theory (e. g. Lurie, lect 15, theorem 9).

Notice that if it is, then the canonical map

$MU_\bullet(X) \otimes_{MU_\bullet} E_\bullet \stackrel{}{\longrightarrow} E_\bullet(X)$

is an equivalence. (For $E =$ KU this was originally proven in Conner-Floyd 66.) In this form the statement has generalizations beyond complex orientation. See at cobordism theory determining homology theory.

## Properties of Landweber-exact spectra

### Phantom maps

Between Landweber exact spectra, every phantom map is already null-homotopic. (e.g. Lurie lect 17, corollary 7).

## References

For the special case of $E =$KU the statement is originally due to

• P. Conner, E. Floyd, The relation of cobordism to K-theories, Lecture Notes in Mathematics 28, 1966 pdf

The general result for complex orientation originates in

• Peter Landweber, Homological properties of comodules over $MU^\ast(MU)$ and $BP^\ast(BP)$, American Journal of Mathematics (1976): 591-610.

Lecture notes include