Chromatic Homotopy and Telescopic Localization

1 MU and \(\cM _{fg}\)

A ring spectrum \(R\) is

complex oriented if it is a equipped with a ring map from the complex bordism spectrum \(MU \to R\). Such a map provides the cohomology theory \(R^*\) with Chern classes for complex vector bundles satisfying a Whitney sum formula.

Given two line bundles \(L_1,L_2:X \to BU(1)\), there is a universal formula for the Chern class of their tensor product:

\[ c_1(L_1\otimes L_2) = F_R(c_1(L_1),c_1(L_2))\]

\(F_R\) is a power series in two variables with coefficients in \(R^*\), and encodes the structure of a (1-dimensional commutative) formal group law. A formal group law is an abelian group structure on the formal \(R\)-scheme \(\Spf (R[[x]])\). More concretely, this means that \(F_R\) satisfies group axioms, such as associativity: \(F_R(x,F_R(y,z)) = F_R(F_R(x,y),z)\).

An important result of Quillen says that \(MU\), the universal complex oriented ring spectrum, has the universal formal group law. In particular, \(MU_* = L\), where \(L\) is the Lazard ring, defined by the universal property \(\Hom (L,R) = FGL(R)\), where \(FGL\) is the set of formal group laws on \(R\).

However, the connection between \(MU\) and formal group laws doesn’t stop there. Recall we have the Adams-Novikov spectral sequence:

\[E_2 = \Ext _{MU_*MU}(MU_*,MU_*X) \implies \pi _* X\]

The \(E_2\) term can be interpreted in terms of formal groups (which are formal group schemes Zariski-locally isomorphic to a formal group law). The \(\Ext \) in the spectral sequence is taken in the category of comodules over the (graded) Hopf algebroid \((MU_*,MU_*MU)\). However, this Hopf algebroid presents \(\cM _{fg}\), the moduli stack of formal groups. \(\cM _{fg}\) has a line bundle \(\omega \) that is the Lie algebra of the universal formal group. Then the Adams-Novikov \(E_2\) term can be reinterpreted as

\[E_2 = H^*(\cM _{fg};(MU_*X)_{\text {even/odd}}\otimes \omega ^{\otimes *}) \implies \pi _* X\]

Where we treat the even and odd degree parts of \(MU_*X\) as a quasicoherent sheaf on \(\cM _{fg}\).

Thus via \(MU\), stable homotopy is tied to formal groups.

The study of formal groups simplifies a bit when localized at a prime. \((\cM _{fg})_{(p)}\) has a simpler presentation as a graded Hopf algebroid \((BP_*,BP_*BP)\), where \(BP_*\) is the ring \(\ZZ _{(p)}[v_1,v_2,\dots ]\), with \(|v_i| = 2(p^i-1)\). In fact this Hopf algebroid (as suggested by the notation) comes from a ring spectrum called \(BP\). In fact, \(MU_{(p)}\) decomposes into summands that are shifts of \(BP\).

\((\cM _{fg})_{(p)}\) is a well understood stack. We can draw a picture of its points \(\Spc ((\cM _{fg})_{(p)})\)

0	1	2	3	\(\cdots \)	\(\infty \)
\(\cdot \)	\(\cdot \)	\(\cdot \)	\(\cdot \)	\(\cdots \)	\(\cdot \)

There is one for each natural number and a point \(\infty \). One way to interpret each point is that it classified a formal group over an algebraically closed field up to isomorphism. The point \(n\) corresponds to a height \(n\) formal group. For example, when \(p=3\), a height \(n\) formal group is one such that if we choose coordinates so that the formal group is defined by a power series \(F\), then \(F(x,F(x,x)) = ux^{3^n} + \dots \) where \(u\) is a unit and \(\dots \) indicates higher order terms.

The point \(0\) classifies a formal group in characteristic \(0\), and the rest of the points classify formal groups in characteristic \(p\).

Another way to interpret the picture is that it classifies invariant prime ideals of \(BP_*\) in the Hopf algebroid \((BP_*,BP_*BP)\). The point \(n\) corresponds to the ideal \((v_0,v_1,\dots ,v_{n-1})\) where \(v_0 = p\).

The space also has a topology, where the open sets are the intervals from \(0\) to \(n\). In particular, specialization increases height.