Quantum phase estimation algorithm

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.

Statement

The coefficient ring of complex cobordism is $M{U}_{*}(*)=M{U}_{*}\cong \mathbb{\mathbb{Z}}[x_1,x_2,\dots ]$, where the degree of $x_i$ is 2i. This is isomorphic to the graded Lazard ring $L_{*}$. 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 $L_{*}\to R_{*}$. 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 $R_{*}$. Define for a topological space X

$E_{*}(X)=M{U}_{*}(X){\otimes }_{M{U}_{*}}{R}_{*}$

Here $R_{*}$ gets its $M{U}_{*}$-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 $R_{*}$ is flat over $M{U}_{*}$, 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 $v_1,v_2,\cdots \in M{U}_{*}$ such that we have the following: Suppose that $M_{*}$ is a graded $M{U}_{*}$-module and the sequence $(p,v_1,v_2,\dots ,v_n)$ is regular for M, for every p and n. Then

$E_{*}(X)=M{U}_{*}(X){\otimes }_{M{U}_{*}}{M}_{*}$

is a homology theory on CW-complexes.

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

Remarks

• There is also a version for Brown–Peterson cohomology BP. The spectrum BP is a direct summand of $M{U}_{(p)}$ with coefficients ${\mathbb{\mathbb{Z}}}_{(p)}[v_1,v_2,\dots ]$. 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 $B{P}_{*}$ which are invariant under coaction of $B{P}_{*}BP$ are the $I_n=(p,v_1,\dots ,v_n)$. This allows to check flatness only against the $B{P}_{*}/I_n$ (see Landweber, 1976).

• The LEFT can be strengthened as follows: let ${\mathcal{ℰ}}_{*}$ be the (homotopy) category of Landweber exact $M{U}_{*}$-modules and $\mathcal{ℰ}$ the category of MU-module spectra M such that ${\pi }_{*}M$ is Landweber exact. Then the functor ${\pi }_{*}\mathcal{ℰ}\to {\mathcal{ℰ}}_{*}$ is an equivalence of categories. The inverse functor (given by the LEFT) takes $M{U}_{*}$-algebras to (homotopy) MU-algebra spectra (see Hovey, Strickland, 1999, Thm 2.7).

Examples

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 $M{U}_{*}\to K_{*}$ is also known as the Todd genus. We have then an isomorphism

$K_{*}(X)=M{U}_{*}(X){\otimes }_{M{U}_{*}}{K}_{*},$

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

Modern reformulation

A module M over $M{U}_{*}$ is the same as a quasi-coherent sheaf $\mathcal{ℱ}$ over $\text{Spec }L$, where L is the Lazard ring. If $M=M{U}_{*}(X)$, then M has the extra datum of a $M{U}_{*}MU$ coaction. A coaction on the ring level corresponds to that $\mathcal{ℱ}$ is an equivariant sheaf with respect to an action of an affine group scheme G. It is a theorem of Quillen that $G\cong \mathbb{\mathbb{Z}}[b_1,b_2,\dots ]$ and assigns to every ring R the group of power series

$g(t)=t+b_1{t}^2+b_2{t}^3+\cdots \in R[[t]]$.

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

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

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

Refinements to $E_{\mathrm{\infty }}$-ring spectra

While the LEFT is known to produce (homotopy) ring spectra out of $M{U}_{*}$, it is a much more delicate question to understand when these spectra are actually $E_{\mathrm{\infty }}$-ring spectra. As of 2010, the best progress was made by Jacob Lurie. If X is an algebraic stack and $X\to {\mathcal{ℳ}}_{fg}$ 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 $X\to M_p(n)$ is etale, then this presheaf can be refined to a sheaf of $E_{\mathrm{\infty }}$-ring spectra (see Goerss). This theorem is important for the construction of topological modular forms.

References

• P. Goerss, Realizing families of Landweber exact homology theories
• Hovey, Mark and Strickland, Neil P., Morava K-theories and localisation, Mem.Amer. Math. Soc., 139 (1999), no. 666.
• P. S. Landweber, Homological properties of comodules over $MU*(MU)$ and $BP*(BP)$, American Journal of Mathematics 98 (1976), 591–610.
• J. Lurie, Chromatic Homotopy Theory, Lecture Notes (2010)