\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*{spectral symmetric algebra}

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

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

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

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{spectral_affine_lines}{Spectral affine lines}\dotfill \pageref*{spectral_affine_lines} \linebreak
\noindent\hyperlink{spectral_superpoints}{Spectral superpoints}\dotfill \pageref*{spectral_superpoints} \linebreak
\noindent\hyperlink{free_algebras}{Free $\mathbb{E}_\infty$-algebras}\dotfill \pageref*{free_algebras} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a [[ring spectrum]] $R$ and [[module spectrum]] $E$ over $R$, there is the [[free construction|free]] $R$-[[algebra spectrum]] induced by $E$, this is the \emph{symmetric algebra} spectrum $Sym_R E$, in direct analogy to the construction of [[symmetric algebras]] on [[vector spaces]] (e.g. \hyperlink{Khan16}{Khan 16}).

The underlying spectrum of the symmetric algebra is simply the [[direct sum]] ([[wedge sum]] of spectra) of all the $n$-fold [[smash products]] of $E$ over $R$ [[homotopy quotient|homotopy quotiented]] by the canonical [[∞-action]] of the [[symmetric group]] $\Sigma(n)$ (by [[permutation]] of factors):

\begin{displaymath}
Sym_R E 
  \;\coloneqq\;
  \underset{n \in \mathbb{N}}{\vee}
  (E^{\wedge^n})/\Sigma(n)
  \,.
\end{displaymath}
In contrast to ordinary [[symmetric algebras]] on ordinary [[modules]] over ordinary [[rings]], this means that the symmetric algebra on $R$ itself, regarded as a [[module spectrum]] itself has interesting structure:

\begin{displaymath}
\begin{aligned}
    Sym_R R
    & \simeq
    \underset{n \in \mathbb{N}}{\vee} 
     R / \Sigma(n)
    \\
    & \simeq
    \underset{n \in \mathbb{N}}{\vee} R \wedge (B \Sigma(n))_+
    \\
    & \simeq
    R \wedge
    \left(
      \underset{n \in \mathbb{N}}{\coprod}
      B \Sigma(n)
    \right)_+
  \end{aligned}
  \,,
\end{displaymath}
where $B \Sigma_n$ denotes the [[homotopy type]] of the [[classifying space]] of the [[symmetric group]] on $n$ elements (given for instance by the [[topological space]] $Emb(\{1, \cdots, n\}, \mathbb{R}^{\infty})/\Sigma(n)$).

In particular for $R = \mathbb{S}$ the [[sphere spectrum]], then the \emph{absolute [[spectral affine line]]} $Sym_{\mathbb{S}}\mathbb{S}$ is nontrivial. This, and its structure of a [[Hopf ring spectrum]], is discussed in \hyperlink{StricklandTurner97}{Strickland-Turner 97}.

More generally,

\begin{displaymath}
R\{x_1, \cdots, x_n\} \coloneqq Sym_R (R^{\vee^n})
\end{displaymath}
is like a [[polynomial ring|polynomial]] [[algebra spectrum]] over $R$. Beware that there is a similar construction that is in general different, namely

\begin{displaymath}
R[x_1, \cdots, x_n] \coloneqq R \wedge \Sigma^\infty(\mathbb{N}_+)
\end{displaymath}
where on the right we have the ``monoid $\infty$-algebra'' of the [[natural numbers]], directly analogous to the [[∞-group ∞-ring]] construction. There is a canonical comparison homomorphisms

\begin{displaymath}
R\{x_1, \cdots, x_n\} \longrightarrow R[x_1, \cdots, x_n]
  \,.
\end{displaymath}
This is an [[equivalence]] if $R$ is of [[characteristic zero]] (\hyperlink{Khan16}{Khan 16, prop. 2.7.4}).

 Similarly, if $E = \Sigma^n R \simeq R \wedge S^n$ is the $n$-fold [[suspension]] of $R$, regarded as an $R$-[[module spectrum]], then

\begin{displaymath}
\begin{aligned}
    Sym_R(\Sigma^n R)
    & \simeq
    R \wedge \left(
      \underset{k \in \mathbb{N}}{\coprod} S^{n k}/\Sigma(k)
    \right)_+
    \\
    & \simeq 
    R \wedge \left(
      \underset{k \in \mathbb{N}}{\coprod}
      (B \Sigma(k))^{n \tau_k}
    \right)
  \end{aligned}
  \,,
\end{displaymath}
where on the right we have the [[Thom space]] of the [[vector bundle]] $\tau_n$ [[associated bundle|associated]] to the $\Sigma(n)$-[[universal principal bundle]] via the canonical [[action]] of $\Sigma(n)$ on $\mathbb{R}^n$ (see also at \href{permutation#ClassifyingSpaceAndThomSpace}{symmetric group -- Classifying space and Thom space}).

The operation $Sym_R$ is of course functorial, and hence any choice of $R$-linear map $f_x \colon R \to E$ induces morphisms

\begin{displaymath}
R \wedge B \Sigma(n)_+ \simeq Sym^n_R(R) \longrightarrow Sym_R^n(E)
  \,.
\end{displaymath}
These correspond to [[power operation]] in [[generalized (Eilenberg-Steenrod) cohomology]] (\hyperlink{Rezk10}{Rezk 10, slide 4}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{spectral_affine_lines}{}\subsubsection*{{Spectral affine lines}}\label{spectral_affine_lines}

For $R$ a ([[connective spectrum|connective]]) [[E-∞ ring]], the

\begin{displaymath}
\mathbb{A}^1_R \coloneqq Spec( Sym_R(R) )
\end{displaymath}
is the [[spectral affine line]] over $R$.

\hypertarget{spectral_superpoints}{}\subsubsection*{{Spectral superpoints}}\label{spectral_superpoints}

By the discussion at \emph{[[spectral super scheme]]}, it makes good sense to regard the concept of a [[spectral scheme]] over an [[even cohomology theory|even]] [[periodic ring spectrum]] $R$ as the lift to [[spectral geometry]] of the concept of ordinary [[super schemes]]. Under this identification the ordinary [[superpoint]], which is the [[spectrum of a commutative ring]] of the graded symmetric algebra on a single odd generator (``[[ring of dual numbers]]'')

\begin{displaymath}
\mathbb{A}_k^{0 \vert 1}
  \;\simeq\;
  Spec( Sym_k (k[1]) )
\end{displaymath}
is given by the corresponding [[spectral scheme]] given by the spectral symmetric algebra on the [[suspension spectrum]] of $R$:

\begin{displaymath}
\begin{aligned}
    R^{0 \vert 1}
     &\coloneqq
    Spec
    \left(
      Sym_R (\Sigma R)
    \right)
    \\
    & \simeq
    Spec\left(
      R \wedge Sym_{\mathbb{S}}(\Sigma \mathbb{S})
    \right)
  \end{aligned}
\end{displaymath}
\hypertarget{free_algebras}{}\subsubsection*{{Free $\mathbb{E}_\infty$-algebras}}\label{free_algebras}

The free $\mathbb{E}_\infty$-algebra over $R$ on $n$ generators is the spectrum $Sym_R(R^{\vee n})$. This is typically denoted $R\{x_1, \ldots, x_n\}$. If $R$ is connective, $\pi_0(R\{x_1, \ldots, x_n\})$ can be identified with the polynomial algebra $(\pi_0 R)[x_1, \ldots, x_n]$. The spectrum $R\{x_1, \ldots, x_n\}$ satisfies the following universal property: for any other $\mathbb{E}_\infty$-$R$-algebra $T$,

\begin{displaymath}
Map_{Alg_R^{\mathbb{E}_\infty}}(R\{x_1, \ldots, x_n\}, T) \simeq (\Omega^\infty T)^{\times n}.
\end{displaymath}
See (\hyperlink{Lurie18}{Lurie 2018, Notation B.1.1.2})

\hypertarget{references}{}\subsection*{{References}}\label{references}

The symmetric algebra spectrum of the [[sphere spectrum]], and its structure as a [[Hopf ring spectrum]] is discussed in

\begin{itemize}%
\item [[Neil Strickland]], [[Paul Turner]], \emph{Rational Morava $E$-theory and $D S^0$}, Topology Volume 36, Issue 1, January 1997, Pages 137-151 (\href{http://hopf.math.purdue.edu/Strickland-PTurner/rme.pdf}{pdf})

\end{itemize}
Symmetric algebras in the context of [[power operation]] on [[generalized (Eilenberg-Steenrod) cohomology]] are discussed in

\begin{itemize}%
\item [[Charles Rezk]], \emph{Power operations in Morava E-theory -- a survey} (2009) (\href{http://www.math.uiuc.edu/~rezk/midwest-2009-power-ops.pdf}{pdf})


\item [[Charles Rezk]] \emph{Power operations in Morava $E$-Theory} (2010) (\href{http://www.math.uiuc.edu/~rezk/baltimore-2010-power-ops-handout.pdf}{pdf})


\item [[Charles Rezk]], \emph{Isogenies, power operations, and homotopy theory}, article (\href{http://www.math.uiuc.edu/~rezk/rezk-icm-talk-posted.pdf}{pdf}) and talk at ICM 2014 (\href{http://www.math.uiuc.edu/~rezk/rezk-icm-2014-slides.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item [[Jacob Lurie]], examples 3.5.4 in \emph{Elliptic Cohomology I} (\href{http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf}{pdf})


\item [[Jacob Lurie]], notation B.1.1.2 in \emph{Spectral Algebraic Geometry} (\href{http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf}{pdf})


\item [[Adeel Khan]], sections 2.6 and 2.7 of \emph{Brave new motivic homotopy theory I} (\href{https://arxiv.org/abs/1610.06871}{arXiv:1610.06871})



\end{itemize}
[[!redirects spectral symmetric algebra]]

[[!redirects symmetric algebra spectra]]

[[!redirects symmetric algebra spectrum]]

[[!redirects spectral polynomial algebra]] [[!redirects spectral polynomial algebras]]

[[!redirects spectral polynomial ring]] [[!redirects spectral polynomial rings]]



\end{document}