\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 - derived group schemes and (pre-)orientations}

This is a sub-entry of

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

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

This entry contains a basic introduction to derived [[group scheme]]s and their orientations.

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]]


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



\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]]


\item [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]



\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{derived_group_schemes}{Derived group schemes}\dotfill \pageref*{derived_group_schemes} \linebreak
\noindent\hyperlink{preorientations}{Preorientations}\dotfill \pageref*{preorientations} \linebreak
\noindent\hyperlink{orientations}{Orientations}\dotfill \pageref*{orientations} \linebreak
\noindent\hyperlink{the_multiplicative_derived_group_scheme}{The Multiplicative Derived Group Scheme}\dotfill \pageref*{the_multiplicative_derived_group_scheme} \linebreak
\noindent\hyperlink{preorientations_of_}{Preorientations of $G_m$}\dotfill \pageref*{preorientations_of_} \linebreak
\noindent\hyperlink{orientations_of_}{Orientations of $G_m$}\dotfill \pageref*{orientations_of_} \linebreak
\noindent\hyperlink{connection_to_complex_orientation}{Connection to complex orientation}\dotfill \pageref*{connection_to_complex_orientation} \linebreak
\noindent\hyperlink{the_additive_derived_group_scheme}{The Additive Derived Group Scheme}\dotfill \pageref*{the_additive_derived_group_scheme} \linebreak


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

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

\begin{displaymath}
A_{S^1}({*}) \to A(\mathbb{C}P^\infty) = 
  A^{Bor}_{S^1}({*}).
\end{displaymath}
The problem we (begin) to address here is how to extend this equivariant cohomology to other spaces besides the point. This requires derived algebraic geometry.

\hypertarget{derived_group_schemes}{}\section*{{Derived group schemes}}\label{derived_group_schemes}

Recall that a commutative [[group scheme]] over a scheme $X$ is a functor

\begin{displaymath}
G: \mathrm{Sch} /X^{op} \to \mathrm{Ab}
\end{displaymath}
such that composition with the forgetful functor $F: \mathrm{Ab} \to \mathrm{Set}$ is representable.

We would like extend this definition to the world of [[derived scheme]]s. There are two problems

\begin{enumerate}%
\item Because of the higher categorical nature of derived schemes Hom sets are spaces.


\item Everything should in the $\infty$-setting, that is defined only up to homotopy.



\end{enumerate}
We will not worry about the second concern and address the first by replacing the category $\mathrm{Ab}$ with $\mathrm{TopAb}$ and $\mathrm{Set}$ with $\mathrm{Top}$.

\begin{quote}%
The following definition is somewhat restrictive and really should incorporate more of the $\infty$-structure.


\end{quote}
\textbf{Definition} A commutative [[derived group scheme]] over a [[derived scheme]] $X$ is a topological functor

\begin{displaymath}
G: \mathrm{DSch} / X^{op} \to \mathrm{TopAb}
\end{displaymath}
such that composition with the forgetful functor $F: \mathrm{TopAb} \to \mathrm{Top}$ is representable (up to weak equivalence) by an object which is [[flat object|flat]] over $X$.

\textbf{Examples}

\begin{enumerate}%
\item Let $X$ be a scheme, then we have an associated [[derived scheme]] $\overline{X}$. The structure sheaf of $\overline{X}$ is obtained by viewing the structure sheaf of $X$ as a presheaf of $E_\infty$-rings and then sheafifying. We then have an equivalence between commutative [[derived group scheme]]s over $\overline{X}$ and commutative [[group scheme]]s which are flat over $X$.


\item For $X$ a [[derived scheme]] we have a map from commutative [[derived group scheme]]s over $X$ to commutative [[group scheme]]s which are flat over $\pi_0 X$.



\end{enumerate}
\hypertarget{preorientations}{}\section*{{Preorientations}}\label{preorientations}

Throughout $A$ will be an $E_\infty$-ring, $X$ the affine [[derived scheme]] $\mathrm{Spec} A$, $G$ a commutative [[derived group scheme]] over $X$, $A_{S^1}$ the $E_\infty$-ring given by $\Gamma (G)$, and $A^{\mathbb{C}P^\infty}$ the $E_\infty$-ring given by

\begin{displaymath}
A^{\mathbb{C}P^\infty} = \mathrm{Hom}_{E_\infty} (\mathbb{C}P^\infty , A).
\end{displaymath}
\textbf{Definition}(Preliminary) A preorientation of $G$ is a morphism of commutative [[derived group scheme]]s over $X$

\begin{displaymath}
\sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to G,
\end{displaymath}
where $\mathrm{Spf} A^{\mathbb{C}P^\infty}$ is the completion wrt the ideal $\mathrm{ker} (A^{\mathbb{C}P^\infty} \to A^{pt} = A)$. A preorientation is an orientation if the induced map

\begin{displaymath}
\hat \sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to \hat G
\end{displaymath}
is an isomorphism.

Suppose that $G$ is affine, then a map

\begin{displaymath}
\sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to G
\end{displaymath}
corresponds to a map $A_{S^1} \to A^{\mathbb{C}P^\infty}$ which is the same as a map

\begin{displaymath}
\mathbb{C}P^\infty \to \mathrm{Hom} (X, G) = G(X).
\end{displaymath}
Hence we are led to the following definition.

\textbf{Definition} Let $X$ be a [[derived scheme]] and $G$ a commutative [[derived group scheme]] over $X$. A preorientation of $G$ is a morphism of topological commutative monoids

\begin{displaymath}
\mathbb{P} (\mathbb{C} [\alpha]) = \mathbb{C}P^\infty \to G(X).
\end{displaymath}
Notice that $\mathbb{C}P^\infty$ is nearly freely generated. Indeed it follows from the fundamental theorem of algebra that as a topological monoid $\mathbb{C}P^\infty$ is generated by $\mathbb{C}P^1$ subject to the single relation that $\mathrm{pt} = \mathbb{C}P^0 \subset \mathbb{C}P^1$ is the monoidal unit.

\textbf{Proposition} A preorientation up to homotopy is a map

\begin{displaymath}
S^2 \simeq \mathbb{C}P^1 \to G(X)
\end{displaymath}
that is an element of $\pi_2 (G(X))$.

Hence, we always have at least one preorientation: the trivial one which corresponds to $0 \in \pi_2 (G(X))$.

\hypertarget{orientations}{}\section*{{Orientations}}\label{orientations}

As motivation recall that a map $s: A \to B$ of 1-dim [[formal group]]s is an isomorphism if and only if $s'$ is invertible. We would like to encode this in our derived language (without defining $s'$).

\textbf{Definition} Let $A$ be an $E_\infty$-ring, $G$ a commutative [[derived group scheme]] over $\mathrm{Spec} A$ and $\sigma : S^2 \to G(A)$ a preorientation. Then $\sigma$ is an orientation if

\begin{enumerate}%
\item $\pi_0 G \to \mathrm{Spec} \pi_0 A$ is smooth of relative dimension 1, and


\item The map induced by $\beta : \omega \to \pi_2 A$

\begin{displaymath}
\pi_n A \otimes_{\pi_0 A} \omega \to \pi_{n+2} A
\end{displaymath}
is an isomorphism for each $n$.



\end{enumerate}
Note that (2) implies that $A$ is weakly periodic. Conversely, if $A$ is weakly periodic then (2) is equivalent to $\beta$ being an isomorphism.

Before defining $\beta$ and $\omega$ we extend the above definition to [[derived group scheme]]s over an arbitrary [[derived scheme]].

\textbf{Definition} Let $X$ be a [[derived scheme]], $G$ a commutative [[derived group scheme]] over $X$ and $\sigma: S^2 \to G(X)$ a preorientation. Then $\sigma$ is an orientation if $(G, \sigma) |_U$ is an orientation for all $U \subset X$ affine.

We now define the module $\omega$ and the map $\beta$. Let $A$ be an $E_\infty$-ring, $G$ a commutative [[derived group scheme]] over $\mathrm{Spec} A$ and let $G_0 = \pi_0 G$ a scheme over $\mathrm{Spec} \pi_0 A$. Let $\Omega$ denote the sheaf of differentials on $G_0 / \pi_0 A$. Then define

\begin{displaymath}
\omega := i^* \Omega, \; i: \mathrm{Spec} \pi_0 A \to G
\end{displaymath}
is the identity section.

Now let $U \hookrightarrow G$ be an open affine subscheme containing the identity section, so $U = \mathrm{Spec} B$ for some $E_\infty$-ring $B$. Then $\sigma$ induces a map $B \to A^{S^2}$ which is the same as a map

\begin{displaymath}
\pi_0 B \to A(S^2)= \pi_0 A \oplus \pi_2 A .
\end{displaymath}
It is a fact that the map $\pi_0 B \to \pi_2 A$ is a derivation over $\pi_0 A$ and hence has a classifying map which yields a map

\begin{displaymath}
\beta : \omega \to \pi_2 A .
\end{displaymath}
\hypertarget{the_multiplicative_derived_group_scheme}{}\section*{{The Multiplicative Derived Group Scheme}}\label{the_multiplicative_derived_group_scheme}

The naive guess for $G_m$ is $GL_1$, where $GL_1 (A) = A^x = (\pi_0 A)^x$. It is true that $GL_1$ is a [[derived scheme]] over $\mathrm{Spec} \mathbf{S}$, however it is not flat, nor is $GL_1 (A)$ an Abelian group as $A$ is an $E_\infty$-ring and not an honestly commutative ring.

If $A$ is rational, that is there is a map $H \mathbb{Q} \to A$, then $GL_1 (A)$ can be given an Abelian group structure. Hence, $GL_1$ is a perfectly good [[group scheme]] defined on the category of rational $E_\infty$-rings, however this category is too small; there are too few rational $E_\infty$-rings.

Recall that for a ring $R$, $R^x = \mathrm{Hom}_R (R [ t, t^{-1} ] , R)$. Further, recall that for a group $M$ we can form the group algebra $R[M]$ which is really a [[Hopf algebra]]. Then $\mathrm{Spec} R[M]$ is a [[group scheme]] over $\mathrm{Spec} A$. Further, $R[m]$ is characterized by

\begin{displaymath}
\mathrm{Ring} ( R[M] , B) = \mathrm{Ring} (R, B) \times \mathrm{Mon} (M,B)
\end{displaymath}
where $\mathrm{Mon}$ is the category of monoids and the ring $B$ is thought of as a monoid wrt to multiplication. Motivated by these observations we make the following definitions.

\textbf{Definition} Let $A$ be an $E_\infty$-ring and $M$ a topological Abelian monoid, then we can define $A[M]$ which is characterized by

\begin{displaymath}
\mathrm{Alg}_A (A[M], B) \simeq \mathrm{Alg}_A (A,B) \times \mathrm{TopMon} (M, B^\times ).
\end{displaymath}
Recall that because of the higher categorical nature of things, the hom-sets above are spaces and the symbol $\simeq$ indicates weak equivalence of spaces.

\textbf{Definition} Let $A$ be an $E_\infty$-ring. We define the multiplicative group corresponding to $A$ as

\begin{displaymath}
G_m = \mathrm{Spec} A[ \mathbb{Z}].
\end{displaymath}
$G_m$ is a derived commutative group scheme over $\mathrm{Spec} A$.

Note that $\pi_* ( A[ \mathbb{Z}]) = (\pi_* A) [ \mathbb{Z}]$. Also, the map $\pi_0 G_m \to \pi_0 \mathrm{Spec}  A$ is smooth of relative dimension 1.

\hypertarget{preorientations_of_}{}\subsection*{{Preorientations of $G_m$}}\label{preorientations_of_}

\textbf{Proposition} For any $E_\infty$-ring $A$, we have a bijection (of sets) between preorientations of $G_m$ and maps $\mathbf{S} [ \mathbb{C} P^\infty ] \to A$.

The proof follows from the fact that $\mathbf{S}$ is initial in the category of $E_\infty$-rings and the mapping property of $A [ \mathbb{Z}]$.

\textbf{Corollary} $\mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ]$ is the moduli space of preorientations of $G_m$. That is, if $G_m$ is defined over $\mathrm{Spec} A$, then a preorientation of $G_m$ is the same as a map $\mathrm{Spec} A \to \mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ]$.

\hypertarget{orientations_of_}{}\subsection*{{Orientations of $G_m$}}\label{orientations_of_}

We consider the map $\beta : \omega \to \pi_2 A$ where

\begin{displaymath}
\omega = i^* \Omega^1_{\pi_0 G / \pi_0 \mathrm{Spec} A}
\end{displaymath}
and $i$ is the identity section. Note that $\pi_0 G_m = (\pi_0 A ) [t, t^{-1}]$, hence it follows that $\omega$ is canonically trivial, so an orientation is just an element $\beta_\sigma \in \pi_2 A$ such that $\beta_\sigma$ is invertible in $\pi_* A$.

Let $\beta$ denote the (universal) orientation of $\mathbf{S} [ \mathbb{C} P^\infty]$. Then we have the following.

\textbf{Theorem} $\mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty] [ \beta^{-1}]$ is the moduli space of orientations of $G_m$.

It is a theorem of Snaith, that this moduli space has the homotopy type of $KU$ the spectrum of complex K-theory. Note that by considering the homtopy fixed points of a certain action there is a way to recover $KO$ as well.

\hypertarget{connection_to_complex_orientation}{}\subsection*{{Connection to complex orientation}}\label{connection_to_complex_orientation}

Let $A$ be an $E_\infty$-ring, so in particular $A$ defines a cohomology theory. An orientation of $G_m$ over $\mathrm{Spec} A$ is a map $KU \to A$. A complex orientation of $A$ is a map $MU \to A$. Recalling that $KU$ is complex oriented, we see that an orientation of $G_m$ gives a complex orientation by precomposing with the map $MU \to KU$.

\hypertarget{the_additive_derived_group_scheme}{}\section*{{The Additive Derived Group Scheme}}\label{the_additive_derived_group_scheme}

The naive definition of $G_a$ is $\mathbf{A}^1$, where $\mathbf{A}^1 (A)$ is the additive group of $A$. It is true that $\mathbf{A}^1$ is a [[derived scheme]] over $\mathrm{Spec} \mathbf{S}$, however it is not flat as for an $E_\infty$-ring $A$

\begin{displaymath}
\pi_k \mathbf{A}^1_A = \oplus_{n \ge 0} A^{-k} (B \Sigma_n )
\end{displaymath}
where as if it were flat we would have

\begin{displaymath}
\pi_k \mathbf{A}^1_A = \pi_k A [x] .
\end{displaymath}
Also, $\mathbf{A}^1$ is not commutative. $\mathbf{A}^1 (A)$ is an infinite loop space, but not an Abelian monoid. Again $\mathbf{A}^1$ is a derived group scheme when restricted to rational $E_\infty$-rings.

We no restrict to the category of integral $E_\infty$-rings, i.e. those equipped with a map $H \mathbb{Z} \to A$. Note that in this category $H \mathbb{Z}$ is initial.

\textbf{Definition} For $A$ an integral $E_\infty$-ring define

\begin{displaymath}
G^A_a = \mathrm{Spec} ( A \otimes_\mathbb{Z} \mathbb{Z} [x] ).
\end{displaymath}
It can be shown that $G^A_a$ is flat and has the correct amount of commutativity.

\begin{quote}%
Why can't we just use $\mathrm{Spec} A [ \mathbb{N}]$?


\end{quote}
\textbf{Proposition} For all integral $E_\infty$-rings, preorientations of $G_a^A$ are in bijective correspondence with maps $H \mathbb{Z} [ \mathbb{C} P^\infty] \to A$. Consequently, $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty]$ is the moduli space of preorientations of $G_a$.

Now, $\pi_* H \mathbb{Z} [ \mathbb{C} P^\infty] = H_* (\mathbb{C} P^\infty , \mathbb{Z} )$. The right side is a free divided power series on a generator $\beta$ where $\beta \in \pi_2 H \mathbb{Z} [ \mathbb{C} P^\infty]$.

\textbf{Proposition} $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty ] [ \beta^{-1}]$ is the moduli space of orientations of $G_a$.

\textbf{Proposition} $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty ] [ \beta^{-1}] = KU \otimes \mathbb{Q}$. Hence the Chern character yields an isomorphism with rational periodic cohomology.



\end{document}