A First Look at an Equivariant Elliptic Cohomology

Usually, besides the information preserved by the formal group law of the elliptic curve, we can’t see any information about the elliptic curve when looking at the output of its associated cohomology theory.  The formal group law only* remembers if the curve was singular/supersingular, and the characteristic of its field.

(*There’s also some crazy invocation of the Serre-Tate Theorem going on s.t. operations upstairs between elliptic curve formal group laws occur downstairs between elliptic cohomology theories but that’s for another post.)

A dream is to have a cohomology theory $E^*(-)$ that encodes all of the geometry of the elliptic curve $C$ used to construct it. One way to phrase this more precisely is by asking for a cohomology theory with a property along the lines of $\textbf{E}^*(pt) = C$.

This property is inspired by an analogue to K-theory, which satisfies something like $\textbf{K}_G(pt) = \mathbb{G}_m$, where $\textbf{K}_G(-) := \text{Spec } K_G(-)$.

Turns out that I’m not alone in this desire, such a cohomology theory has been constructed before! Continue reading A First Look at an Equivariant Elliptic Cohomology

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?

Landweber-Ravenel-Stong Construction Flowchart

Here’s a flowchart I made while preparing for an upcoming talk. I fear that it may be hard to follow without being already familiar with the story, but there’s little harm in posting it. Maybe it’ll help someone navigate the literature.

11137755_10205356925075246_137597741_oSome context:  Continue reading Landweber-Ravenel-Stong Construction Flowchart

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 $0 \to \mathbb{Z} \xrightarrow{\times p} \mathbb{Z} \to \mathbb{Z}/p \to 0$). 

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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