Chromatic homotopy + algebraic geometry =?

Question

In Homotopy Theory there is a famous theorem which shows that every cohomology theory satisfying a certain list of axioms is characterized by a formal group law, and that the spectrum associated to the "universal formal group", which is in some sense the "universal cohomology theory" is characterized by a certain spectrum $MU$ which I believe denotes complex cobordism.

In algebraic geometry there are quite a few cohomology theories, defined over the category of $R$-schemes for an abstract ring $R$, which I believe satisfy a similar set of axioms.

I never heard about the notion of a spectrum in algebraic geometry, but I am also not very familiar with motivic homotopy theory. My question is if there is an analogue classification of algebraic cohomology theories using formal groups, and if so, what is it called/where can I read about it?

If indeed there is such a correspondence, is there similarly a universal group law? What cohomology theory does it correspond to? Does it carry all additional structures of all other cohomology theories, such as weight filtrations/Galois representation structure when $R$ is the spectrum of a field?

If there isn't such an analogy, then why is that the case?

Answer 1

I can imagine more than one way of filling in details to make your question precise. There is one way that seems most natural to me, and the resulting version of your question is answered by the following paper of Naumann-Ostvaer-Spitzweck: https://arxiv.org/abs/0806.0274

That answer takes the following form. By Lazard and Quillen, there is a bijection between

• formal group laws $F$ over a fixed commutative ring $R$,
• and ring homomorphisms $MU_* \rightarrow R$.

If a ring homomorphism $MU_* \rightarrow R$ satisfies the Landweber exactness criterion, then the functor $X\mapsto MU_*(X)\otimes_{MU_*} R$, on pointed topological spaces, is a generalized homology theory. This is how you go from a FGL to a generalized homology theory: you turn the FGL into a ring map $MU_* \rightarrow R$, and then you use Landweber exactness to get a generalized homology theory. If the ring map $MU_*\rightarrow R$ isn't Landweber exact, then you're out of luck: the usual general process to build a generalized homology theory won't work, and you'll have to do something more subtle. (In practice this does happen, e.g. with Morava $K$-theories of positive height--they aren't Landweber exact, and you build them out of their FGLs in a different way.)

Put another way, the correspondence you alluded to in your question is the correspondence between

• formal group laws over $R$ with Landweber exact classifying map $MU_* \rightarrow R$,

• and Landweber exact complex oriented cohomology theories with coefficient ring $R$.

Naumann-Ostvaer-Spitzweck prove a motivic version of the Landweber exact functor theorem. They show that, if $MU_*\rightarrow R$ is a Landweber exact ring map, then the functor $X\mapsto MGL_*(X)\otimes_{MU_*}R$ is a motivic generalized homology theory.

The upshot is that, given an FGL $F$ over a commutative ring $R$, if you could have built a classical generalized homology theory from $F$ in the usual way (i.e., using the Landweber exact functor theorem), then you can also build a motivic generalized homology theory from $F$. In that sense, the motivic generalized homology theories that you get from formal group laws are in bijection with the classical generalized homology theories that you get from formal group laws.

Take a look at the paper of Naumann-Ostvaer-Spitzweck for more details (e.g. what ground field they're working over, which I don't remember). Perhaps some of your sub-questions after the main question are addressed there as well, but I haven't looked carefully. You may have to read other papers about $MGL_*$ to get those answers.

P.S. All formal group laws in this answer are one-dimensional!