Notes on One-Parameter Deformations of Cohomology Theories

Thanks to Dr. Lubin for graciously helping me derive and understand one-parameter families of formal group laws, thanks to Eric Peterson for introducing me to Morava’s Forms of K-theory, thanks to Agnes Beaudry for pointing out that there was a neater way to check for Landweber-exactness.

Today, I want to discuss the opposite procedures of deformations and contractions of complex-orientable cohomology theories.

Really, all I want is to illustrate the fact that, with the appropriate combinations of both procedures, we obtain new cohomology theories. So, we’re going to  examine the construction of one-parameter families of cohomology theories via one-parameter deformations of formal group laws — of particular interest is the case where a continuous deformation causes an increase of chromatic height.

Motivation and Story Leading To This Construction

I was playing with the concept of group contraction, and thinking about the construction of elliptic cohomology theories.

I accidentally constructed a model of Morava E-theory of height 2 at the prime 3. (We didn’t realize that it was $E_{(2)}(3)$ at first, we just thought it was some weird cohomology theory.)

I found this kind of enlightening and so I want to show you how I came across it.

 

Morava constructed a family of elliptic cohomology theories by deforming K-theory (well, by deforming the multiplicative group associated to K-theory). His construction can be viewed as a recipe:

  1. start with a point of interest (an algebraic group)
    e.g., $\mathbb{C}^\times$
    e.g., $y^3 = x^3 – x $ over $\mathbb{F}_3$
  2. deform that point (create a family of algebraic groups indexed by one parameter) 
    e.g., $\mathbb{C}^\times/q^{\mathbb{Z}}$ where $q:= e^{2\pi i}$, we vary the norm of $0 \leq |q| < 1$.
    e.g., $y^3 = x^3 + tx^2 – x$ over $\mathbb{F}_3[[t]]$
  3. look at the formal group laws associated to your family, this is still indexed by one parameter (in fact, they can be viewed as ONE formal group law, if you keep the parameter formal)
  4. either apply the Landweber exact functor theorem to the whole family stalkwise (specializing the parameter), or apply the Landweber exact functor theorem to the ONE formal group law (keeping the parameter formal).

Let me say this again, because when I explain this to people they like me to say it twice.

Morava’s deformation method is a recipe which consists of 4 steps:

  1. construct a continuous family of smooth algebraic groups indexed by q
  2. construct a continuous family of formal group laws indexed by q
  3. construct a family indexed (indexed by q) of contra-functors from Top \to AbGrp (“potential” cohomology theories — we don’t know if they are exact yet)
  4. prove that each member of this family of contravariant functors is exact (or treat the family as one functor, keeping the variable formal, and prove that this functor is exact)

Continue reading Notes on One-Parameter Deformations of Cohomology Theories

Spf $E^*[[x]]$: Your walk through a flower garden

Inspired by the extraordinary expository style of Dr. Kazuya Kato, I’ve started reading parts of a (translated) Japanese children’s book when I’m stuck on a tough paper or concept — revisiting the concept with such a dreamlike world in mind usually unfolds an illustrative perspective. A misty world which begs to be put into firm ground via prolonged formal and concrete afterthought.

He embraces that teaching can be poetic and tantalizing, providing not a definition but a deep and creative hint that causes an exploratory shift in perspective, allowing you to walk down the path to the conclusion yourself. I wanted to try to exposit with this philosophy: confusion is expected and encouraged as impetus for reaching understanding. With that in mind, step into your flower garden.


Planted in a line of earth ($\text{Spec }R$)
there are flowers, $C$, whose heads are smooth projective genus 1 curves
with stems that can retract into the ground,
s.t. the flower meets the earth at one point (a marked point).

11270061_10205668389181654_555974065_n Continue reading Spf $E^*[[x]]$: Your walk through a flower garden

Elliptic Curve Formal Group Laws: Philosophy and Derivation

Eine deutsche Übersetzung des folgenden Abschnitts befindet sich hier.

Philosophical Motivation

In the study of groups with topological structure, we commonly replace the global object (the group) with a local object (the infinitesimal group). We play the following game.

  1. Start with a space.
  2. Define a method of adding two points to get a third point of this space (which is associative, unital, commutative, and has inverses).
  3. Derive an infinitesimal group.

Examples: Continue reading Elliptic Curve Formal Group Laws: Philosophy and Derivation

What does the sphere spectrum have to do with formal group laws?

This post assumes that you’re familiar with the definition of a prime ideal, a local ring, $R_{(p)}$, the sphere spectrum, $\mathbb{S}$, and the Lazard ring, $L$.

During a talk Jacob Lurie gave at Harvard this April, he labeled the moduli space of (1-d commutative) formal group laws as $\text{Spec }\mathbb{S}$.

Eric Peterson kindly explained why $\text{Spec } \mathbb{S} \simeq \text{Spec } L$ and I found his answer so lovely that I wish to share (all mistakes are due to me).

Why is Spec L iso to Spec $\mathbb{S}$?

This is part of the story of geometers working with higher algebra asking “what is an ideal of a ring spectrum?” Continue reading What does the sphere spectrum have to do with formal group laws?

The Landweber exact-functor theorem

This post assumes familiarity with formal group laws, the definition of exact sequences, the motivation of the Landweber-Ravenel-Stong construction, that the exactness axioms is one of the generalized Eilenberg-Steenrod axioms, and the fact that formal group laws over $R$ are represented by maps from the Lazard ring to $R$. 

Recall the the Landweber-Ravenel-Stong Construction: $MU^*(X) \otimes_{L} R \simeq E^*(X)$, where $MU^* \simeq L$ and $R \simeq E^*(pt)$.

We know that in general, tensoring with abelian groups does not preserve exact sequences (e.g., applying $-\otimes_{\mathbb{Z}} \mathbb{Z}/2$ to $0 \to \mathbb{Z} \xrightarrow{\times p} \mathbb{Z} \to \mathbb{Z}/p \to 0$). 

11136786_10205364668548828_1871863654_n

So, when does the functor $-\otimes_L R: MU^*(X) \to E^*(X)$ preserve exact sequences?

11157286_10205364655788509_750654639_o Continue reading The Landweber exact-functor theorem