Landweber exact functor theorem in nLab

Contents

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

Conversely, given a formal group FF over a graded ring RR, one can ask whether this arises from some EE in this way.

Now since the Lazard ring LL classifies formal groups, in that any formal group law FF over a ground ring RR is equivalently a ring homomorphism L⟶RL \longrightarrow R, and since moreover Quillen's theorem on MU identifies LL with the cohomology ring of complex oriented cohomology theory L≃MU •(*)L \simeq MU_\bullet(\ast), one may consider the functor

X↦E •(X)≔MU •(X)⊗LR. X \mapsto E_\bullet(X) \coloneqq MU_\bullet(X) \underset{L}{\otimes} R \,.

By construction this is such that E •(*)≃RE_\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 EE.

Landweber exactness is a sufficient condition on RR for E≔MU •(−)⊗ LRE \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 •(X)⊗ MU •E •⟶E •(X) MU_\bullet(X) \otimes_{MU_\bullet} E_\bullet \stackrel{}{\longrightarrow} E_\bullet(X)

is an equivalence. (For E=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).

Examples

• The special case of the Landweber exact functor theorem for complex topological K-theory KU is known as the Conner-Floyd isomorphism.

• Lazard's theorem

• universal complex orientation on MU

• Quillen's theorem on MU

• Landweber-Novikov theorem

References

For the special case of E=E = KU the statement (Conner-Floyd isomorphism) is originally due to:

• Pierre Conner, Edwin Floyd, Theorem 10.1 in: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

The general result for complex oriented cohomology is due to:

• Peter Landweber, Homological properties of comodules over MU *(MU)MU^\ast(MU) and BP *(BP)BP^\ast(BP), American Journal of Mathematics (1976): 591-610 (jstor:2373808)

Textbook accounts:

• Dai Tamaki, Akira Kono, Theorem 6.37 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

Lecture notes:

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 15 Flat modules over ℳ FG\mathcal{M}_{FG} (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 16 The Landweber exact functor theorem (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 17 Phantom maps (pdf)

See also

• Wikipedia, Landweber exact functor theorem

For the even 2-periodic case:

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 , Lecture 18 Even periodic cohomology theories (pdf)