\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{A Survey of Elliptic Cohomology - equivariant cohomology}

This is a sub-entry of

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

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

This entry considers [[equivariant cohomology]] from the perspective of algebraic geometry.

Previous:

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


\item [[A Survey of Elliptic Cohomology - formal groups and cohomology]]


\item [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]


\item [[A Survey of Elliptic Cohomology - elliptic curves]]



\end{itemize}
Next:

\begin{itemize}%
\item [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]


\item [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]



\end{itemize}
\begin{quote}%
the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention, meaning: \textbf{somebody should go through this and polish}


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{introduction}{Introduction}\dotfill \pageref*{introduction} \linebreak
\noindent\hyperlink{notions_of_equivariant_cohomology}{Notions of equivariant cohomology}\dotfill \pageref*{notions_of_equivariant_cohomology} \linebreak
\noindent\hyperlink{borel_equivariant_cohomology}{Borel equivariant cohomology}\dotfill \pageref*{borel_equivariant_cohomology} \linebreak
\noindent\hyperlink{grothendieck_ring_of_equivariant_vector_bundles}{Grothendieck ring of equivariant vector bundles}\dotfill \pageref*{grothendieck_ring_of_equivariant_vector_bundles} \linebreak
\noindent\hyperlink{algebraic_interpretation}{algebraic interpretation}\dotfill \pageref*{algebraic_interpretation} \linebreak
\noindent\hyperlink{group_schemes}{group schemes}\dotfill \pageref*{group_schemes} \linebreak
\noindent\hyperlink{lesson}{lesson}\dotfill \pageref*{lesson} \linebreak


\hypertarget{introduction}{}\section*{{Introduction}}\label{introduction}

The slogan is: for $A$ a [[cohomology theory]], finding [[equivariant cohomology]] $A$-theory corresponds to finding [[group scheme]] over $A({*})$.

The main example we'll be looking at here is complex [[K-theory]].

\hypertarget{notions_of_equivariant_cohomology}{}\section*{{Notions of equivariant cohomology}}\label{notions_of_equivariant_cohomology}

\hypertarget{borel_equivariant_cohomology}{}\subsection*{{Borel equivariant cohomology}}\label{borel_equivariant_cohomology}

\textbf{Remark} if $A_G$ is an [[equivariant cohomology]] theory and if $Y \to X$ is a $G$-[[principal bundle]] then we want that $A_G(Y) \simeq A(X)$. So whenever we have a $G$-space where the $G$-[[action]] is free enough.

Let $X$ be a $G$-space, form the [[Borel construction]] $\mathcal{E}G \times_G X$ with $\mathcal{E}G \to \mathcal{B}G$ the [[universal principal bundle]].

Then we can \emph{define}

\begin{displaymath}
A_G^{Borel}(X) := A(\mathcal{E}G \times_G X)
  \,.
\end{displaymath}
\begin{quote}%
notice that $\mathcal{E}G \times_G X$ is the realization of the [[action groupoid]] $X//G$. This Borel equivariant cohomology theory is what is discussed currently at the entry [[equivariant cohomology]]. The following will actually define a refinement of the discussion currently at [[equivariant cohomology]].


\end{quote}
\textbf{Problem} If the cohomology theory is given by a geometric model, such as [[topological K-theory]] in terms of [[vector bundle]]s or [[elliptic cohomology]] potentially in a [[geometric model for elliptic cohomology]], then the above notion of equivariant cohomology need not coincide with the cohomology theory given by the equivariant version of these geometric models. In particular, equivariant [[vector bundle]]s are geometric cocycles of equivariant [[K-theory]] $K_G$ and there is a morphism

\begin{displaymath}
K_G(X) \to K_G^{Borel}(X)
\end{displaymath}
but it is not an [[isomorphism]]. Instead, $K_G$ is a \emph{completion} of $K_G^{Bor}$.

Here $K_G(X)$ is the [[Grothendieck group]] of [[equivariant vector bundle]]s over the [[G-space]] $X$ (say $G$ is a compact [[Lie group]]).

\hypertarget{grothendieck_ring_of_equivariant_vector_bundles}{}\subsection*{{Grothendieck ring of equivariant vector bundles}}\label{grothendieck_ring_of_equivariant_vector_bundles}

\textbf{Definition} An [[equivariant vector bundle]] over $X$ is

\begin{itemize}%
\item a [[G-space]] $E$ and $G$-equivariant map $p : E \to X$ such that this is a (complex, here) [[vector bundle]] of finite rank


\item for each $g \in G$ the map $g : E_x \to E_{g x}$ is linear.



\end{itemize}
Morephism are the obvious $G$-equivariant morphisms of [[vector bundle]]s.

\emph{examples*}

\begin{enumerate}%
\item $X$ a trivial $G$-space,then a $G$-equivariant vector bundle is a family of complex representations;


\item for $E \to X$ a [[vector bundle]] the $k$th tensor power of $E$ is a $\Sigma_k$-equivariant vector bundle;


\item if $G$ acts smoothly on $X$ then the complexified [[tangent bundle]] $T X \otimes \mathbb{C} \to X$ is a $G$-equivariant vector bundle.



\end{enumerate}
\textbf{remark} The [[category]] of $G$-[[equivariant vector bundle]], has

\begin{itemize}%
\item [[direct sum]] $\oplus$


\item [[tensor product]] $\otimes$



\end{itemize}
And we can pull back $Vect^G$ $\backslash$to $Vecg^H$ along any group homomorphism $\phi : H \to G$

So we are entitled to say

\textbf{definition} the [[Grothendieck group]] of $Vect^G(X)$ is

\begin{displaymath}
K_G(X) := Groth(E \to X, \oplus)
  \,.
\end{displaymath}
With the remaining [[tensor product]] $\otimes$ this yields a commutative [[ring]].

\textbf{proposition}

\begin{enumerate}%
\item if $X = pt$ then $K_G(X) \simeq Rep(G)$ is the [[representation ring]] of $G$.

\begin{enumerate}%
\item in general, $K_G(X)$ is an algebra over $Rep(G)$.

\end{enumerate}

\item if $G$ [[free action|acts freely]] on $X$, then $K_G(X) \simeq K(X/G)$.



\end{enumerate}
So in particular

\begin{displaymath}
K(\mathcal{E}G \times_G X)
  \simeq
  K_G(\mathcal{E}G \times X)
\end{displaymath}
so we get a map

\begin{displaymath}
\alpha : K_G(X) \to K_G(\mathcal{E}G \times X)
  \simeq
  K(\mathcal{E}G \times_G X)
  := K_G^{Borel}(X)
\end{displaymath}
\textbf{theorem} (Atiyah-Segal) This $\alpha$ induces an [[isomorphism]]

\begin{displaymath}
\hat K_G(X) \simeq K_G^{Borel}(X)
\end{displaymath}
where

\begin{displaymath}
\hat K_G(X) := \lim_\leftarrow K_G(X)/I_G^n K_G(X)
\end{displaymath}
where $I_G = ker(Rep(G) \simeq K_G({*}) \to K_G(\mathcal{E}G) \simeq K(\mathcal{B}G)
  \stackrel{\epsilon}{\to} \mathbb{Z})$

\textbf{consider} $X = {*}$, $G = S^1$, $\mathbb{C}P^\infty \simeq \mathcal{B}S^1$

\begin{displaymath}
\alpha : Rep(G)
  \to 
  K(\mathcal{B}G)
  = K(\mathbb{C}P^\infty)
  \simeq
  \mathbb{Z}[ [ t ] ]
\end{displaymath}
\begin{enumerate}%
\item since $S^1$ is an abelian group, every [[irreducible representation]] is 1-dimensional

\begin{displaymath}
\phi : S^1 \to \mathbb{C}^\times
\end{displaymath}

\item $\chi : S^1 \hookrightarrow \mathbb{C}^times$



\end{enumerate}
\begin{displaymath}
Rep(G) \simeq \mathbb{Z}[\chi, \chi^{-1}]
  \,.
\end{displaymath}
\hypertarget{algebraic_interpretation}{}\subsection*{{algebraic interpretation}}\label{algebraic_interpretation}

\textbf{goal now} find an algebraic interpretation of $\alpha$ such that

\begin{displaymath}
Rep(S^1) = \mathcal{o}_{G_m}
\end{displaymath}
and

\begin{displaymath}
Rep(\mathbb{C}P^\infty) = \mathcal{o}_{\hat G_m}
\end{displaymath}
adopt the [[functor of points]] perspective

\begin{displaymath}
X : CRings \to Set
\end{displaymath}
a [[functor]].

For $A \in$ [[CRing]] get a spectrum

\begin{displaymath}
Spec A : R \mapsto CRing(A,R)
\end{displaymath}
for $X$ a functor, it is an [[affine scheme]] if it is a [[representable functor]] in that there is $A$ with $X \simeq Spec A$.

\textbf{examples}

\begin{enumerate}%
\item $\mathbf{A}^n(R) := R^n$


\item $\hat \mathbf{A}^n(R) := Nil(R)^n$


\item $G_m(R) := R^\times \hookrightarrow \mathbf{A}^1(R)$


\item $\mathbb{P}^n(R) := R^{n+1}/\sim$, where $\sim$ is multiplication by $R^\times$



\end{enumerate}
\textbf{proposition}

$G_m$ is affine.

\textbf{proof} $A = \mathbb{Z}[x,x^{-1}]$, let $u \in Spec(\mathbf{A}(R))$ a map $u A \to R$, then define

\begin{displaymath}
\phi : Spec A \to G_m
\end{displaymath}
by

\begin{displaymath}
u \mapsto u(x)
\end{displaymath}
Conversely, given $v \in G_m(R) = R^\times$, define

\begin{displaymath}
\Psi : G_m \to Spec A
\end{displaymath}
by

\begin{displaymath}
v \mapsto \Psi(v)
\end{displaymath}
with

\begin{displaymath}
\Psi(v)(\sum_k a_k x^k) := \sum_k a_k v^k
\end{displaymath}
\textbf{endofproof}

similarly, $\mathbf{A}^n \simeq Spec \mathbb{Z}[x_1, \cdots, x_n]$

\hypertarget{group_schemes}{}\section*{{group schemes}}\label{group_schemes}

Given a functor $X : CRing \to Set$ define the ring of funtions $\mathcal{o}_X$ as

\begin{displaymath}
\mathcal{o}_X := Hom_{Func(CRing,Set)}(X, \mathbf{A}^1)
\end{displaymath}
in the [[functor category]].

\begin{quote}%
notice notation: this is global sectins of the [[structure sheaf]], not the structure sheaf itself, properly speaking


\end{quote}
we have

\begin{displaymath}
\mathcal{o}_{Spec A}
  \simeq
  A
\end{displaymath}
so that in particular

\begin{displaymath}
\mathcal{o}_{G_m} \simeq \mathbb{Z}[x,x^{-1}]
  \,.
\end{displaymath}
\textbf{definition}

Let $X$ be an [[affine scheme]] and $Y : CRing \to Set$ a [[functor]] with a [[natural transformation]] $p : Y \to X$. A \textbf{system of formal coordinates} is a sequence of maps

\begin{displaymath}
X_i : Y \to \hat \mathbf{A}^1
\end{displaymath}
such that

\begin{displaymath}
a \mapsto (x_1(a), \cdots, x_n(a), p(a))
  \in 
  \hat \mathbf{A}^n \times X
\end{displaymath}
is an [[isomorphism]]. A $Y$ that admits a system of formal coordinates is a \textbf{[[formal scheme]]} over $X$.

\begin{quote}%
\textbf{warning} very restrictive definition. See [[formal scheme]]


\end{quote}
A \textbf{[[formal group]]} $G$ over a [[scheme]] $X$ is a one-dimensional [[formal scheme]] with specified abelian [[group]] structure on each [[fiber]] $p^{-1}\{x\}$.

This means that there is a natural map

\begin{displaymath}
\sigma : G \times_X G \to G
\end{displaymath}
and a natural map $\zeta : X \to G$ which maps $x \mapsto 0 \in p^{-1}\{x\}$.

\textbf{definition (formal multiplicative group)}

define $\hat G_m$ on each $R \in CRing$ by

\begin{displaymath}
\hat G_m(R)
  = 
  \left\{
    1+n | n \in Nil(R)
  \right\}
\end{displaymath}
which is a group under multiplication.

there is an [[isomorphism]] of underlying [[formal scheme]]s

\begin{displaymath}
\hat G_m  \simeq \hat \mathbf{A}^1
\end{displaymath}
We compute $\mathcal{o}_{\hat \mathbf{A}^1} \simeq \mathcal{o}_{\hat G_m}$ in two ways:

\begin{enumerate}%
\item Recall that $\hat \mathbf{A}^1$ can be defined as $Spf \; \mathbb{Z} [t]$, so the global sections of the structure sheaf (which is what we have been calling $\mathcal{o}$) is

\begin{displaymath}
\mathcal{o}_{\hat \mathbf{A}^1} = \lim_\rightarrow \mathbb{Z}[t]/ (t^n) = \mathbb{Z} [[t]] .
\end{displaymath}

\item We can also see this in the functor of points perspective. Consider the functor $\mathrm{Spec} \; \mathbb{Z} [t]/ (t^n)$, then for any ring $R$ \$$\hat \mathbf{A}^1 (R) = \lim_\rightarrow \mathrm{Spec} \; \mathbb{Z} [t]/ (t^n) (R).$\$ By the universal property of colimits we have

\begin{displaymath}
\mathrm{Nat} (\hat \mathbf{A}^1 , \mathbf{A}^1 ) \simeq \lim_\leftarrow \mathrm{Nat} (\mathrm{Spec} \; \mathbb{Z}[t]/ (t^n) , \mathbf{A}^1) \simeq \mathbb{Z} [[t]].
\end{displaymath}


\end{enumerate}
\begin{displaymath}
\mathcal{o}_{\hat G_m} \simeq \mathbb{Z}[ [ t] ]
  \,.
\end{displaymath}
\textbf{recall} we have a morphism $\alpha : Rep(S^1) \to K(\mathbb{C}P^\infty)$

such that

\begin{displaymath}
\mathcal{o}_{G_m}
  \simeq
   \mathbb{Z}[x,x^{-1}]
   \simeq
   Rep(S^1)
  \to 
  K(\mathbb{C}P^\infty) \simeq
  \mathbb{Z}[ [ t ] ]
  \simeq
  \mathcal{o}_{\hat G_m}
\end{displaymath}
is the canonical inclusion

\begin{displaymath}
\mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}
\end{displaymath}
\textbf{exercise}

\begin{displaymath}
x \in \mathbb{Z}[x,x^{-1}] \simeq \mathcal{o}_{G_m}
\end{displaymath}
is the natural transformation $R^\times \to R$

and

\begin{displaymath}
t \in \mathbb{Z}[ [t] ] \simeq
  \mathcal{o}_{\hat G_m}
\end{displaymath}
is the natural transformation

\begin{displaymath}
\{1 + Nil(R)\}
  \to
  Nil(R)
  \to 
  R
\end{displaymath}
so that we get a map

\begin{displaymath}
\mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}
\end{displaymath}
by sending $x$ to $1 + t$, and this corresponds to taking the [[germ]] of functions at $1 \in G_m$

\hypertarget{lesson}{}\section*{{lesson}}\label{lesson}

given $G$ an algebraic group such that the [[formal spectrum]] $Spf A(\mathbb{C}P^\infty)$ is the completion $\hat G$, define $A_{S^1}({*}) := \mathcal{o}_{G}$ then passing to germs gives a completion map

\begin{displaymath}
A_{S^1}({*}) \to A(\mathbb{C}P^\infty) = 
  A^{Bor}_{S^1}({*})
\end{displaymath}


\end{document}