In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology.
It is known that a complex orientation of a homology theory leads to a formal group law.
The Landweber exact functor theorem (or LEFT for short) can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.
The coefficient ring of complex cobordism is
This is isomorphic to the graded Lazard ring
This means that giving a formal group law F (of degree
is equivalent to giving a graded ring morphism
is defined inductively as a power series, by Let now F be a formal group law over a ring
-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.
Peter Landweber found another criterion: In particular, every formal group law F over a ring
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
is Landweber exact, homology with integer coefficients
Furthermore, Morava K-theory K(n) is not Landweber exact.
A coaction on the ring level corresponds to that
is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of Quillen that
and assigns to every ring R the group of power series It acts on the set of formal group laws
via These are just the coordinate changes of formal group laws.
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
The Landweber exactness theorem can then be interpreted as a flatness criterion for
While the LEFT is known to produce (homotopy) ring spectra out of
, it is a much more delicate question to understand when these spectra are actually
As of 2010, the best progress was made by Jacob Lurie.
a flat map of stacks, the discussion above shows that we get a presheaf of (homotopy) ring spectra on X.
(the stack of 1-dimensional p-divisible groups of height n) and the map
is etale, then this presheaf can be refined to a sheaf of
This theorem is important for the construction of topological modular forms.