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\begin{document}

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\section*{symmetric algebra}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{explicit_definition}{Explicit definition}\dotfill \pageref*{explicit_definition} \linebreak
\noindent\hyperlink{for_vector_spaces}{For vector spaces}\dotfill \pageref*{for_vector_spaces} \linebreak
\noindent\hyperlink{in_general}{In general}\dotfill \pageref*{in_general} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{symmetric algebra} $S V$ of a vector space is the [[free object|free]] [[commutative algebra]] over $V$.

This construction generalizes to [[group representations]], [[chain complexes]], [[vector bundles]], [[coherent sheaves]], and indeed objects in any [[symmetric monoidal category|symmetric monoidal]] [[linear category|linear categories]] with enough [[colimit|colimits]], where the tensor product distributes over those colimits (as in a [[2-rig]]).

\hypertarget{explicit_definition}{}\subsection*{{Explicit definition}}\label{explicit_definition}

We begin with the construction for vector spaces and then sketch how to generalize it.

\hypertarget{for_vector_spaces}{}\subsubsection*{{For vector spaces}}\label{for_vector_spaces}

Suppose $V$ is a [[vector space]] over a [[field]] $K$. Then the \textbf{symmetric algebra} $S V$ is generated by the elements of $V$ using these operations:

\begin{itemize}%
\item addition and scalar multiplication
\item an associative binary operation $\cdot$

\end{itemize}
subject to these identities:

\begin{itemize}%
\item the identities necessary for $S V$ to be an [[associative algebra]]
\item the identity $v \cdot w = w \cdot v$ for all $v \in V$.

\end{itemize}
It then follows that $S V$ is a [[graded algebra]] where $S^p V$ is spanned by $p$-fold products, that is, elements of the form

\begin{displaymath}
v_1 \cdot \cdots \cdot v_p
\end{displaymath}
where $v_1, \dots, v_p \in V$. Clearly $S V$ is also [[commutative algebra|commutative]].

The symmetric algebra of $V$ is also denoted $Sym V$. It is also called the \textbf{polynomial algebra}. However we should carefully distinguish between polynomials \emph{in} the elements of $V$, which form the algebra $S V$, and polynomial functions \emph{on} the vector space $V$, which form the algebra $S(V^*)$. In quantum physics, a similar construction for Hilbert spaces is known as the [[Fock space]].

\hypertarget{in_general}{}\subsubsection*{{In general}}\label{in_general}

More generally, suppose $C$ is any [[symmetric monoidal category]] and $V \in C$ is any object. Then we can form the [[tensor power|tensor powers]] $V^{\otimes n}$. If $C$ has countable coproducts we can form the [[coproduct]]

\begin{displaymath}
T V = \bigoplus_{n \ge 0} V^{\otimes n}
\end{displaymath}
(which we write here as a [[direct sum]]), and if the tensor product distributes over these coproducts, $T V$ becomes a [[monoid]] object in $C$, with multiplication given by the obvious maps

\begin{displaymath}
V^{\otimes p} \otimes V^{\otimes q} \to V^{\otimes (p+q)}
\end{displaymath}
This monoid object is called the [[tensor algebra]] of $V$.

The [[symmetric group]] $S_n$ acts on $V^{\otimes n}$, and if $C$ is a [[linear category]] over a [[field]] of [[characteristic]] zero, then we can form the symmetrization map

\begin{displaymath}
p_A : V^{\otimes n} \to V^{\otimes n}
\end{displaymath}
given by

\begin{displaymath}
p_A = \frac{1}{n!} \sum_{\sigma \in S_n} \sigma
\end{displaymath}
This is an [[idempotent]], so if idempotents split in $C$ we can form its [[cokernel]], called the $n$th \textbf{symmetric tensor power} or \textbf{symmetric power} $S^n V$. The coproduct

\begin{displaymath}
S V = \bigoplus_{n \ge 0} S^n V
\end{displaymath}
becomes a monoid object called the \textbf{symmetric algebra} of $V$.

If $C$ is a more general sort of symmetric monoidal category, then we need a different construction of $S^n V$. For example, if $C$ is a symmetric monoidal category with finite colimits, we can simply define $S^n V$ to be the [[coequalizer]] of the action of the symmetric group $S_n$ on $V^{\otimes n}$. And if $C$ also has countable coproducts, we can define

\begin{displaymath}
S V = \coprod_{n \ge 0} S^n V
\end{displaymath}
Then, if the tensor product distributes over these colimits (as in a [[2-rig]]), $S V$ will become a [[internalization|commutative monoid object]] in $C$. In fact, it will be the \emph{free} commutative monoid object on $V$, meaning that any morphism in $C$

\begin{displaymath}
V \to A \, ,
\end{displaymath}
where $A$ is a commutative monoid, factors uniquely as the obvious morphism

\begin{displaymath}
V \to S V
\end{displaymath}
followed by a morphism of commutative monoids

\begin{displaymath}
S V \to A  \, ,
\end{displaymath}
as in this commutative triangle:

\begin{displaymath}
\array {
     &          & S V \\
     & \nearrow &                 & \searrow \\
   V &          & \longrightarrow &          & A
}
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{SymmetricAlgebraInCoChainComplexes}\hypertarget{SymmetricAlgebraInCoChainComplexes}{}
\textbf{(symmetric algebra in [[chain complexes]] is [[differential graded-commutative algebra]])}

Let the ambient [[category]] be the [[category of cochain complexes]] over a [[ground field]] of [[characteristic zero]], regarded as a [[symmetric monoidal category]] via the [[tensor product of chain complexes]].

Then for $(V^\bullet, d)$ a [[cochain complex]], the symmetric algebra

\begin{displaymath}
Sym\left( (V^\bullet, d) \right)
  \;\in\; CMon( Ch^\bullet, \otimes )
\end{displaymath}
is the [[differential graded-commutative algebra]] whose underlying [[graded algebra]] is the [[graded-commutative algebra]] on $V^\bullet$, and whose [[differential]] is the original $d$, extended, uniquely, as a graded [[derivation]] of degree +1.

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{CochainCohomologyOfSymmetricAlgebraOnCochainComplex}\hypertarget{CochainCohomologyOfSymmetricAlgebraOnCochainComplex}{}
Let the ambient [[category]] be the [[category of cochain complexes]] over a [[ground field]] of [[characteristic zero]], regarded as a [[symmetric monoidal category]] via the [[tensor product of chain complexes]] and consider the [[differential graded-commutative algebra]] $Sym(V^\bullet,d)$ free on a cochain complex $(V^\bullet,d)$ from example \ref{SymmetricAlgebraInCoChainComplexes}.

Then the [[cochain cohomology]] (of the [[forgetful functor|underlying]] cochain complex) of $Sym(V^\bullet,d)$ is the graded symmetric algebra on the [[cochain cohomology]] of $(V^\bullet,d)$:

\begin{displaymath}
H^\bullet\left( 
    Sym\left(V^\bullet,d\right)
  \right)
  \;\simeq\;
  Sym\left( H^\bullet(V^\bullet,d) \right)
  \,.
\end{displaymath}
\end{prop}
See also \href{https://mathoverflow.net/a/91243/381}{this MO discussion}.

\begin{proof}
We have the following sequence of [[linear isomorphisms]]:

\begin{displaymath}
\begin{aligned}
    H^\bullet
    \left(
      Sym( V^\bullet,d )
    \right)
    & \simeq
    H^\bullet
    \left(
      \underset{k \in \mathbb{N}}{\oplus}
      \left( 
        (V^\bullet,d)^{\otimes_k} 
      \right)^{\Sigma_k}
    \right)
    \\
    & \simeq
    \underset{k \in \mathbb{N}}{\oplus}
    H^\bullet\left(
       \left( (V^\bullet,d)^{\otimes_k}\right)^{\Sigma_k}
    \right)
    \\
    & \simeq
    \underset{k \in \mathbb{N}}{\oplus}
    H^\bullet\left(
       \left( (V^\bullet,d)^{\otimes_k}\right)
    \right)^{\Sigma_k}
    \\
    & \simeq
    \underset{k \in \mathbb{N}}{\oplus}
    \left(
      H^\bullet\left(
        V^\bullet,d
      \right)^{\otimes_k}
    \right)^{\Sigma_k}
    \\
    & =
    Sym\left( H^\bullet(V^\bullet,d)\right)
  \end{aligned}
\end{displaymath}
Here:

\begin{enumerate}%
\item the first step uses that, while \emph{a priori} the symmetric algebra is equivalently the [[quotient]] of the [[tensor algebra]] by the [[symmetric group]] [[action]], in [[characteristic zero]] this is equivalently [[invariants]] of the [[symmetric group]] [[action]], because here $V^G \to V \to V_G$ is a [[linear isomorphism]];


\item the second step uses that [[cochain cohomology]] respects [[direct sums]];


\item the third step uses that for [[finite groups]] in [[characteristic zero]], taking [[invariants]] is compatible with passing to [[cochain cohomology]] (\href{invariant#TakingInvariantsForFiniteGroupCommutesWithTaingHomologyInCharZero}{this prop.}):


\item the fourth step is the [[Künneth theorem]] for [[ordinary homology]] over a [[field]] (\href{K&#252;nneth+theorem#InordinaryHomology}{this prop.}).



\end{enumerate}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[symmetric algebra spectrum]]

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

\begin{itemize}%
\item [[Stacks Project]], \href{http://stacks.math.columbia.edu/tag/00DM}{Tag 00DM}.


\item [[Bourbaki]], \emph{Alg\`e{}bre}, chap. III, \S{} 6.



\end{itemize}
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}
[[!redirects symmetric algebras]] [[!redirects symmetric power]] [[!redirects symmetric powers]] [[!redirects symmetric tensor power]] [[!redirects symmetric tensor powers]]

[[!redirects symmetric product]] [[!redirects symmetric products]]

[[!redirects symmetric tensor algebra]] [[!redirects symmetric tensor algebras]]



\end{document}