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\section*{A Survey of Elliptic Cohomology - formal groups and cohomology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

This is a sub-entry of

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology]]

\end{itemize}
see there for background and context.

This entry disscusses basics of [[formal group law]]s arising from [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory|cohomology theories]]

Previous:

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology - cohomology theories]]

\end{itemize}
Next:

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

\end{itemize}
\hypertarget{rough_notes_from_a_talk}{}\subsection*{{rough notes from a talk}}\label{rough_notes_from_a_talk}

\begin{quote}%
the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs \textbf{somebody to go through it and polish it}


\end{quote}
\textbf{Formal groups and elliptic cohomology.}

In all of the following, all [[cohomology theory|cohomology theories]] are [[multiplicative cohomology theory|multiplicative]] and all [[formal group law]]s are one-dimensional (and commutative).

\textbf{[[A Survey of Elliptic Cohomology - cohomology theories|Last time]].} we saw that orienting a [[periodic cohomology theory|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.)

\textbf{Today} we discuss a generalization of the above statement: orienting a [[weakly periodic cohomology theory|weakly periodic]] [[even cohomology theory]] $A$ gives a [[formal group]] over $A^0(\bullet)$. In particular, [[elliptic cohomology]] theories give [[elliptic curve]]s over $A^0(\bullet)$.

\hypertarget{formal_group_laws_and_landwebers_criterion}{}\subsection*{{Formal group laws and Landweber's criterion}}\label{formal_group_laws_and_landwebers_criterion}

[[formal group law|Formal group law]]s of dimension $1$ over $R$ are classified by [[morphism]]s 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 cohomology theory|complex cobordism]], in particular $MP(\bullet)$ is isomorphic to [[Lazard ring|Lazard's ring]].

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

\textbf{[[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.

\textbf{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]] $R$.

\textbf{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.

\hypertarget{formal_groups_from_formal_group_laws}{}\subsection*{{Formal groups from formal group laws}}\label{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 $a+_f b=f(a,b)$.

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

\textbf{Example.} Let $A$ be an oriented complex oriented cohomology theory. Then computing [[Chern class]]es of [[line bundle]]s is the same as evaluating the formal group law of $A$ on some algebra. Recall that [[line bundle]]s on $X$ are [[classifying space|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.

\textbf{Question:} When do two formal group laws yield isomorphic groups?

\textbf{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{\tt \symbol{126}}$\phi$ is zero.) Hence formal group laws form a category.

\textbf{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 $\prod_kH^{n+2k}(X,\mathbb{Q})$.

\textbf{Formal groups.} A [[formal group]] is a group in the [[category]] of [[formal scheme]]s.

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

\textbf{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 $X$.)

\textbf{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 \emph{algebraizable}.

\textbf{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]]

\begin{displaymath}
O_{\mathrm{Spf} R}(\mathrm{Spf} R)=\lim_n (R/I^n)
  \,.
\end{displaymath}
(This is incomplete description, one needs to talk sheaves of ideals instead)

\hypertarget{formal_group_laws_from_elliptic_curve}{}\subsubsection*{{formal group laws from elliptic curve}}\label{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

\begin{displaymath}
\itexarray{
    Spf R[[x,y]] \simeq
    Spf[[x]] \times Spf R[[y]]
    &\to& Spf R[[z]]
    \\
    f(x,y)&\leftarrow |& z 
  }
\end{displaymath}
Isomorphic [[formal group law]]s give isomorphism|isomorphic] (of [[formal group]]s) 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 group]]s from [[elliptic curve]]s over $R$

\textbf{Definition} An \textbf{[[elliptic curve]]} over a commutative [[ring]] $R$ is a [[group object]] in the [[category]] of [[scheme]]s over $R$ that is a relative 1-dimensional, , [[smooth scheme|smooth curve]], [[proper scheme|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]] $\sigma_0$

\begin{displaymath}
E \to Spec(R) \stackrel{\sigma_0}{\to} E
\end{displaymath}
(the one dual to the map that maps everything to $0 \in R$), we get a [[ringed space]] $(\hat E, \hat O_{E,0})$

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

then

\begin{displaymath}
\hat O_{E \times E, (0,0)} \simeq
  \hat O_{E,0}
  \hat \otimes_k
  \hat O_{E,0}
  \simeq 
  k[[x,y]]
\end{displaymath}
\textbf{example} \textbf{(Jacobi quartics)}

\begin{displaymath}
y^2 = 1- 2 \delta x^2 + \epsilon x^4
\end{displaymath}
defines $E$ over $R = \mathbb{Z}[Y_Z,\epsilon, \delta]$.

The corresponding [[formal group law]] is \textbf{Euler's formal group law}

\begin{displaymath}
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}
\end{displaymath}
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 [[integral cohomology]] and [[K-theory]], respectively).

\hypertarget{weakly_periodic_cohomology_theories_and_formal_groups}{}\subsection*{{weakly periodic cohomology theories and formal groups}}\label{weakly_periodic_cohomology_theories_and_formal_groups}

A [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A$ is \textbf{[[weakly periodic cohomology theory|weakly periodic]]} if the natural map

\begin{displaymath}
A^2({*}) \otimes_{A^0({*})} A^n({*}) \stackrel{\simeq}{\to}
  A^{n+2}({*})
\end{displaymath}
is an [[isomorphism]] for all $n \in \mathbb{Z}$.

Compare with the notion of a [[periodic cohomology theory]].

\hypertarget{relation_to_formal_groups}{}\section*{{Relation to formal groups}}\label{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 cohomology theory|weakly periodic]], [[even cohomology theory|even]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A^\bullet$, we look at the induced map on $A^\bullet$ from a morphism

\begin{displaymath}
i_0 : {*} \to \mathbb{C}P^\infty
\end{displaymath}
and take the kernel

\begin{displaymath}
J := ker(i_0^* : A^0(\mathbb{C}P^\infty) \to A^0({*}))
\end{displaymath}
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]]

\begin{displaymath}
Lie(Spf A^0(\mathbb{C}P^\infty))
  \simeq
  ker(i_0^*)/ker(i_0^*)^2
\end{displaymath}
is a free $A^0({*})$-module, we can pick a generator $t$ and this gives an [[isomorphism]]

\begin{displaymath}
Spf(A^0(\mathbb{C}P^\infty)) \simeq Spf(A^0({*})[[t]])
\end{displaymath}
if $A^0(\mathbb{C}P^\infty) A^0({*})[ [t] ]$ then $i_0^*$ ``forgets the $t$-coordinate''.

\textbf{Definition} An \textbf{elliptic cohomology theory} over $R$ is

\begin{itemize}%
\item a commutative [[ring]] $R$


\item an [[elliptic curve]] $E/R$


\item a [[weakly periodic cohomology theory|weakly periodic]], [[multiplicative cohomology theory|multiplicative]], [[even cohomology theory|even]] [[cohomology theory]] $A^\bullet$


\item [[isomorphism]]s $A^0({*}) \simeq R$ and $\hat E \simeq Spf(A^0(\mathbb{C}P^\infty))$.



\end{itemize}
So we have on one side

\begin{displaymath}
\itexarray{
    \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)
  }
\end{displaymath}
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.

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


\end{document}