Landweber exact functor theorem

The coefficient ring of complex cobordism is $MU\_*(*)\; =\; MU\_*\; \backslash cong\; \backslash Z[x\_1,x\_2,\backslash dots]$, where the degree of $x\_i$ is $2i$. This is isomorphic to the graded Lazard ring $\backslash mathcal\{\}L\_*$. This means that giving a formal group law F (of degree $-2$) over a graded ring $R\_*$ is equivalent to giving a graded ring morphism $L\_*\backslash to\; R\_*$. Multiplication by an integer $n>0$ is defined inductively as a power series, by

:$[n+1]^F\; x\; =\; F(x,\; [n]^F\; x)$ and $[1]^F\; x\; =\; x.$

Let now F be a formal group law over a ring $\backslash mathcal\{\}R\_*$. Define for a topological space X

:$E\_*(X)\; =\; MU\_*(X)\backslash otimes\_\{MU\_*\}R\_*$

Here $R\_*$ gets its $MU\_*$-algebra structure via F. The question is: is E a homology theory? It is obviously a homotopy invariant functor, which fulfills excision. The problem is that tensoring in general does not preserve exact sequences. One could demand that $R\_*$ be flat over $MU\_*$, but that would be too strong in practice. Peter Landweber found another criterion:

:Theorem (Landweber exact functor theorem)

: For every prime p, there are elements $v\_1,v\_2,\backslash dots\; \backslash in\; MU\_*$ such that we have the following: Suppose that $M\_*$ is a graded $MU\_*$-module and the sequence $(p,v\_1,v\_2,\backslash dots,\; v\_n)$ is regular for $M$, for every p and n. Then

::$E\_*(X)\; =\; MU\_*(X)\backslash otimes\_\{MU\_*\}M\_*$

:is a homology theory on CW-complexes.

In particular, every formal group law F over a ring $R$ yields a module over $\backslash mathcal\{\}MU\_*$ since we get via F a ring morphism $MU\_*\backslash to\; R$.

• There is also a version for Brown–Peterson cohomology BP. The spectrum BP is a direct summand of $MU\_\{(p)\}$ with coefficients $\backslash Z\_\{(p)\}[v\_1,v\_2,\backslash dots]$. The statement of the LEFT stays true if one fixes a prime p and substitutes BP for MU.
• The classical proof of the LEFT uses the Landweber–Morava invariant ideal theorem: the only prime ideals of $BP\_*$ which are invariant under coaction of $BP\_*BP$ are the $I\_n\; =\; (p,v\_1,\backslash dots,\; v\_n)$. This allows to check flatness only against the $BP\_*/I\_n$ (see Landweber, 1976).
• The LEFT can be strengthened as follows: let $\backslash mathcal\{E\}\_*$ be the (homotopy) category of Landweber exact $MU\_*$-modules and $\backslash mathcal\{E\}$ the category of MU-module spectra M such that $\backslash pi\_*M$ is Landweber exact. Then the functor $\backslash pi\_*\backslash colon\backslash mathcal\{E\}\backslash to\; \backslash mathcal\{E\}\_*$ is an equivalence of categories. The inverse functor (given by the LEFT) takes $\backslash mathcal\{\}MU\_*$-algebras to (homotopy) MU-algebra spectra (see Hovey, Strickland, 1999, Thm 2.7).

The archetypical and first known (non-trivial) example is complex K-theory K. Complex K-theory is complex oriented and has as formal group law $x+y+xy$. The corresponding morphism $MU\_*\backslash to\; K\_*$ is also known as the Todd genus. We have then an isomorphism

: $K\_*(X)\; =\; MU\_*(X)\backslash otimes\_\{MU\_*\}K\_*,$

called the Conner–Floyd isomorphism.

While complex K-theory was constructed before by geometric means, many homology theories were first constructed via the Landweber exact functor theorem. This includes elliptic homology, the Johnson–Wilson theories $E(n)$ and the Lubin–Tate spectra $E\_n$.

While homology with rational coefficients $H\backslash mathbb\{Q\}$ is Landweber exact, homology with integer coefficients $H\backslash mathbb\{Z\}$ is not Landweber exact. Furthermore, Morava K-theory K(n) is not Landweber exact.

A module M over $\backslash mathcal\{\}MU\_*$ is the same as a quasi-coherent sheaf $\backslash mathcal\{F\}$ over $\backslash text\{Spec\; \}L$, where L is the Lazard ring. If $M\; =\; \backslash mathcal\{\}MU\_*(X)$, then M has the extra datum of a $\backslash mathcal\{\}MU\_*MU$ coaction. A coaction on the ring level corresponds to that $\backslash mathcal\{F\}$ is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of Quillen that $G\; \backslash cong\; \backslash Z[b\_1,\; b\_2,\backslash dots]$ and assigns to every ring R the group of power series

:$g(t)\; =\; t+b\_1t^2+b\_2t^3+\backslash cdots\backslash in\; Rt$.

It acts on the set of formal group laws $\backslash text\{Spec\; \}L(R)$ via

:$F(x,y)\; \backslash mapsto\; gF(g^\{-1\}x,\; g^\{-1\}y)$.

These are just the coordinate changes of formal group laws. Therefore, one can identify the stack quotient $\backslash text\{Spec\; \}L\; //\; G$ with the stack of (1-dimensional) formal groups $\backslash mathcal\{M\}\_\{fg\}$ and $M\; =\; MU\_*(X)$ defines a quasi-coherent sheaf over this stack. Now it is quite easy to see that it suffices that M defines a quasi-coherent sheaf $\backslash mathcal\{F\}$ which is flat over $\backslash mathcal\{M\}\_\{fg\}$ in order that $MU\_*(X)\backslash otimes\_\{MU\_*\}M$ is a homology theory. The Landweber exactness theorem can then be interpreted as a flatness criterion for $\backslash mathcal\{M\}\_\{fg\}$ (see Lurie 2010).

While the LEFT is known to produce (homotopy) ring spectra out of $\backslash mathcal\{\}MU\_*$, it is a much more delicate question to understand when these spectra are actually $E\_\backslash infty$-ring spectra. As of 2010, the best progress was made by Jacob Lurie. If X is an algebraic stack and $X\backslash to\; \backslash mathcal\{M\}\_\{fg\}$ a flat map of stacks, the discussion above shows that we get a presheaf of (homotopy) ring spectra on X. If this map factors over $M\_p(n)$ (the stack of 1-dimensional p-divisible groups of height n) and the map $X\backslash to\; M\_p(n)$ is etale, then this presheaf can be refined to a sheaf of $E\_\backslash infty$-ring spectra (see Goerss). This theorem is important for the construction of topological modular forms.

• Chromatic homotopy theory

• {{cite web|first=Paul|last= Goerss| url=http://www.math.northwestern.edu/~pgoerss/papers/banff.pdf|title= Realizing families of Landweber exact homology theories}}
• {{citation|last1=Hovey|first1= Mark|last2= Strickland|first2= Neil P.|url=http://math.wesleyan.edu/~mhovey/papers/kn.ps|archive-url=https://web.archive.org/web/20041207115433/http://math.wesleyan.edu/~mhovey/papers/kn.ps|url-status=dead|archive-date=2004-12-07|title= Morava K-theories and localisation| journal= Memoirs of the American Mathematical Society | volume= 139 | year=1999|issue= 666|mr=1601906|doi=10.1090/memo/0666|url-access=subscription}}
• {{cite journal|first=Peter S.|last= Landweber|authorlink=Peter Landweber|title=Homological properties of comodules over $MU\_*(MU)$ and $BP\_*(BP)$|journal= American Journal of Mathematics | volume= 98 |issue= 3|year=1976|pages= 591–610|jstor= 2373808|doi= 10.2307/2373808}}.
• {{cite web|first=Jacob|last= Lurie|authorlink=Jacob Lurie| url=http://www.math.harvard.edu/~lurie/252x.html|title= Chromatic Homotopy Theory. Lecture Notes |year=2010}}

Algebraic Topology