nLab A Survey of Elliptic Cohomology - formal groups and cohomology

This is a sub-entry of

• A Survey of Elliptic Cohomology

see there for background and context.

This entry disscusses basics of formal group laws arising from periodic multiplicative cohomology theories

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rough notes from a talk

the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs somebody to go through it and polish it

Formal groups and elliptic cohomology.

In all of the following, all cohomology theories are multiplicative and all formal group laws are one-dimensional (and commutative).

Last time. we saw that orienting a periodic even cohomology theory gives a formal group law over the cohomology ring $A^0(\bullet)$. (Note: $A^0$ and not $A^\bullet$ because of the periodicity property.)

Today we discuss a generalization of the above statement: orienting a weakly periodic even cohomology theory $A$ gives a formal group over $A^0(\bullet)$. In particular, elliptic cohomology theories give elliptic curves over $A^0(\bullet)$.

Formal group laws and Landweber’s criterion

Formal group laws of dimension $1$ over $R$ are classified by morphisms from the Lazard ring to $R$.

We can define $A_f^n(X)=MP^n(X)\otimes_{MP(\bullet)}R$. Here $MP$ denotes complex cobordism, in particular $MP(\bullet)$ is isomorphic to Lazard's ring.

Definition. A sequence $v_0,\ldots,v_n$ of elements of $R$ is regular if endomorphisms of $R/(v_0,\ldots,v_{k-1})$ given by multiplication by $v_k$ are injective for all $0\le k\le n$.

Landweber criterion Let $f(x,y)$ be a formal group law and $p$ a prime, $v_i$ the coefficient of $x^{p^i}$ in $[p]_f(x)=x+_f\cdots+_fx$. If $v_0,\ldots,v_i$ form a regular sequence for all $p$ and $i$ then $f(x,y)$ gives a cohomology theory via the formula with tensor product above.

Example. $g_a(x,y)=x+y$, $[p]_a(x)=px$, $v_0=p$, $v_i=0$ for all $i\ge1$; regularity condtions imply that the zero map $R/(p)\to R/(p)$ must be injective. The last statement implies that $R$ contains the rational numbers as a subring.

Note that $HP^*(X,R)=\prod_k H^{n+2k}(X,R)$ is a cohomology theory over any ring $R$.

Example. $g_m(x,y)=xy$, $[p]_m(x)=(x+1)^p-1$, $v_0=p$, $v_1=1$, $v_i=0$ for all $i \gt 1$. The regularity conditions are trivial. Hence we know that $K^*(X)=MP^*(X)\otimes_{MP(\bullet)} \mathbb{Z}$ is a cohomology theory.

Formal groups from formal group laws

Given a commutative topological $R$-algebra $A$ and a formal group law $f(x,y)$ if $f(a,b)$ converges for all $a,b\in A$ and the formula giving an inverse to $a$ converges for all $a\in A$, we get an abelian group $(A,+_f)$, where $a+_f b=f(a,b)$.

Example. For any $A$ the pair $(N(A),+_f)$ is an abelian group, where $N(A)$ denotes the set of nilpotent elements of $A$.

Example. Let $A$ be an oriented complex oriented cohomology theory. Then computing Chern classes of line bundles is the same as evaluating the formal group law of $A$ on some algebra. Recall that line bundles on $X$ are classified by maps from $X$ to $\mathbb{C}P^\infty$, pairs of line bundles are classified by maps to $\mathbb{C}P^\infty \times \mathbb{C}P^\infty$, and tensor product of line bundles gives a map $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$. Now apply cohomology functor to the sequence $X\to \mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$. We have a degree 0 element $t$ in the cohomology of $\mathbb{C}P^\infty$. Its image in the cohomology of $\mathbb{C}P^\infty \times \mathbb{C}P^\infty$ is a formal group law. The image of this formal group law in the cohomology of $X$ makes sense if $X$ is a finite cell complex so that $A^0(X)$ is a nilpotent algebra.

Question: When do two formal group laws yield isomorphic groups?

Definiton. A homomorphism of formal group laws $f$ and $g$ over $A$ is a formal power series $\phi\in A[x]$ such that $\phi(f(x,y))=g(\phi(x),\phi(y))$. (The constant term of~$\phi$ is zero.) Hence formal group laws form a category.

Example. If $R$ contains rational numbers as a subring, then we have two canonical homomorphisms. The first one is $\exp\colon g_a\to g_m$, where $\exp(x)=\sum_{k \gt 0}x^k/k!$. Its inverse is $\log\colon g_m\to g_a$, where $\log(x)=\sum_{k \gt 0}(-1)^{k+1}x^k/k$. This shows up in cohomology as Chern character. (Isomorphism from $K^n(X)\otimes_{\mathbb{Z}} \mathbb{Q}$ to $\prod_kH^{n+2k}(X,\mathbb{Q})$.

Formal groups. A formal group is a group in the category of formal schemes.

A formal scheme $\hat{Y}$ is defined for any closed immersion of schemes $Y \hookrightarrow X$. Intuitively the formal scheme $\hat Y$ is the $\infty$-jet bundle in the normal direction of $Y$ inside of $X$.

Definition. The locally ringed space $\hat Y$ is defined as the topological space $Y$ with structure sheaf $lim O_X/{\mathcal{I}}^n$, where $\mathcal{I}$ is the defining sheaf of ideals of the closed immersion $Y\hookrightarrow X$. (Where $Y$ is a closed subscheme of $X$.)

Examples. $X=\hat Y$ when $Y=X$. $\mathrm{Spec} k[t]=X$, $Y=V(t)$, $\hat X=k[t,t^{-1}]$.

In fact not every locally noetherian formal scheme can be obtained as a completion of a single noetherian scheme in another scheme; such formal schemes are called algebraizable.

Definition. (formal spectrum) The formal spectrum $\mathrm{Spf} R$ of a commutative noetherian ring $R$ with a specified ideal $I \subset R$ whose powers define a local basis of a topology around $0$ which is Hausdorff, is the locally ringed space with the underlying topological space $\mathrm{Spec} R/I$ whose global sections of the structure sheaf are the limit

$$O_{\mathrm{Spf} R}(\mathrm{Spf} R)=\lim_n (R/I^n) \,.$$

(This is incomplete description, one needs to talk sheaves of ideals instead)

formal group laws from elliptic curve

Recall from the above that a given a formal group law $F(x,y) \in R[ [x,y] ]$ we get te structure of a formal group on the formal spectrum $Spf$ by taking the product to be given by

$$\array{ Spf R[[x,y]] \simeq Spf[[x]] \times Spf R[[y]] &\to& Spf R[[z]] \\ f(x,y)&\leftarrow |& z }$$

Isomorphic formal group laws give [isomorphism|isomorphic]] (of formal groups) if $G$ a formal group has $G \simeq Spf R[ [z] ]$; we must choose such an iso to get a formal group law.

Now we get formal groups from elliptic curves over $R$

Definition An elliptic curve over a commutative ring $R$ is a group object in the category of schemes over $R$ that is a relative 1-dimensional, , smooth curve, proper curve over $R$.

This implies that it has genus 1. (by a direct argument of the Chern class of the tangent bundle.)

Given an elliptic curve over $R$, $E \to Spec R$, we get a formal group $\hat E$ by completing $D$ along its identity section $\sigma_0$

$$E \to Spec(R) \stackrel{\sigma_0}{\to} E$$

(the one dual to the map that maps everything to $0 \in R$), we get a ringed space $(\hat E, \hat O_{E,0})$

example if $R$ is a field $k$, then the structure sheaf $\hat O_{E,0} \simeq k[ [z] ]$

then

$$\hat O_{E \times E, (0,0)} \simeq \hat O_{E,0} \hat \otimes_k \hat O_{E,0} \simeq k[[x,y]]$$

example (Jacobi quartics)

$$y^2 = 1- 2 \delta x^2 + \epsilon x^4$$

defines $E$ over $R = \mathbb{Z}[Y_Z,\epsilon, \delta]$.

The corresponding formal group law is Euler’s formal group law

$$f(x,y) = \frac{x\sqrt{1- 2 \delta y^2 + \epsilon y^4} + y \sqrt{1- 2 \delta x^2 + \epsilon x^4}} {1- \epsilon x^2 y^2}$$

if $\Delta := \epsilon(\delta^2 - \epsilon)^2 \neq 0$ then this is a non-trivial elliptic curve.

If $\Delta = 0$ then $f(x,y) \simeq G_m, G_a$ (additive or multiplicative formal group law corresponding to ordinary cohomology and topological K-theory KU, respectively).

weakly periodic cohomology theories and formal groups

A multiplicative cohomology theory $A$ is weakly periodic if the natural map

$$A^2({*}) \otimes_{A^0({*})} A^n({*}) \stackrel{\simeq}{\to} A^{n+2}({*})$$

is an isomorphism for all $n \in \mathbb{Z}$.

Compare with the notion of a periodic cohomology theory.

Relation to formal groups

One reason why weakly periodic cohomology theories are of interest is that their cohomology ring over the space $\mathbb{C}P^\infty$ defines a formal group.

To get a formal group from a weakly periodic, even multiplicative cohomology theory $A^\bullet$, we look at the induced map on $A^\bullet$ from a morphism

$$i_0 : {*} \to \mathbb{C}P^\infty$$

and take the kernel

$$J := ker(i_0^* : A^0(\mathbb{C}P^\infty) \to A^0({*}))$$

to be the ideal that we complete along to define the formal scheme $Spf A^0(\mathbb{C}P^\infty)$ (see there for details).

Notice that the map from the point is unique only up to homotopy, so accordingly there are lots of chocies here, which however all lead to the same result.

The fact that $A$ is weakly periodic allows to reconstruct the cohomology theory essentially from this formal scheme.

To get a formal group law from this we proceed as follows: if the Lie algebra $Lie(Spf A^0(\mathbb{C}P^\infty))$ of the formal group

$$Lie(Spf A^0(\mathbb{C}P^\infty)) \simeq ker(i_0^*)/ker(i_0^*)^2$$

is a free $A^0({*})$-module, we can pick a generator $t$ and this gives an isomorphism

$$Spf(A^0(\mathbb{C}P^\infty)) \simeq Spf(A^0({*})[[t]])$$

if $A^0(\mathbb{C}P^\infty) A^0({*})[ [t] ]$ then $i_0^*$ “forgets the $t$-coordinate”.

Definition An elliptic cohomology theory over $R$ is

• a commutative ring $R$

• an elliptic curve $E/R$

• a weakly periodic, multiplicative, even cohomology theory $A^\bullet$

• isomorphisms $A^0({*}) \simeq R$ and $\hat E \simeq Spf(A^0(\mathbb{C}P^\infty))$.

So we have on one side

$$\array{ \hat E &\stackrel{\simeq}{\to}& Spf A^0(\mathbb{C}P^\infty) \\ \downarrow && \downarrow \\ Spec R &\stackrel{\simeq}{\to}& Spec A^0({*}) \\ \downarrow^{\sigma_0} && \downarrow \\ \hat E &\stackrel{\simeq}{\to}& Spf A^0(\mathbb{C}P^\infty) }$$

We can check that the Landweber exactness criterion is satisfied for the formal group law of the Jacobi quartic, i.e. for Euler's formal group law? over $\mathbb{Z}[\Delta^{-1}, \epsilon, \delta, 1/2]$, so this provides an example of an elliptic cohomology theory.

$$A^n_G(X) = M P (X) \otimes_{M P({*})} \mathbb{Z}[\Delta^{-1}, \epsilon, \delta, 1/2]$$