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\begin{document}

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\section*{semifree dga}

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

\hypertarget{differentialgraded_objects}{}\paragraph*{{Differential-graded objects}}\label{differentialgraded_objects}

[[!include differential graded objects - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{roiters_theorem}{Roiter's theorem}\dotfill \pageref*{roiters_theorem} \linebreak
\noindent\hyperlink{relation_to_lie_algebroids}{Relation to Lie $\infty$-algebroids}\dotfill \pageref*{relation_to_lie_algebroids} \linebreak
\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A [[differential graded algebra]] is \textbf{semifree} (or semi-free) if the underlying [[graded algebra]] is free: if after forgetting the differential, it is isomorphic as a graded algebra to a (polynomial) [[tensor algebra]] of some ([[super vector space|super]])[[graded vector space]].

A [[differential graded-commutative algebra]] is \textbf{semifree} (or semi-free) if the underlying graded-commutative algebra is free: if after forgetting the differential, it is isomorphic as a graded-commutative algebra to a [[Grassmann algebra]] of some [[graded vector space]] .

\hypertarget{roiters_theorem}{}\subsection*{{Roiter's theorem}}\label{roiters_theorem}

Roiter's theorem

\begin{itemize}%
\item A. V. Roiter, \emph{Matrix problems and representations of BOCS's}; in Lec. Notes. Math. 831, 288--324 (1980)

\end{itemize}
says: semi-free differential graded algebras are in bijective correspondence with [[coring]]s with a [[grouplike element]]:

to an $A$-coring $(C,\Delta, A)$ with a grouplike element $g$ associate its [[Amitsur complex]] with underlying graded module $T_A(\Omega^1 A)=\oplus_{n=0}^\infty (\Omega^1 A)^{\otimes_A n}$ where $\Omega^1=ker\,\epsilon$ and differential linearly extending the formulas $d a = g a - a g$ for $a\in A$ and

\begin{displaymath}
d c = g\otimes c + (-1)^n c\otimes g +\sum_{i=1}^n (-1)^i c_1\otimes\ldots\otimes c_{i-1}\otimes\Delta(c_i)\otimes c_{i+1}\otimes\ldots\otimes c_n
\end{displaymath}
for $c=c_1\otimes_A\ldots\otimes_A c_n\in (ker\,\epsilon)^{\otimes_A n}$;

conversely, to a semi-free dga $\Omega^\bullet A$ one associates the $A$-coring $A g\oplus\Omega^1 A$ where $g$ isa new group-like indeterminate; this is by definition a direct sum of left $A$-modules with a right $A$-module structure given by

\begin{displaymath}
(a g +\omega)a' := a a' g + a d a'+\omega a'.
\end{displaymath}
In other words, we want the commutator $[g,a']=d\omega'$. We obtain an $A$-bimodule. The coproduct on $Ag\oplus\Omega^1 A$ is $\Delta(a g)=a g\otimes g$ and $\Delta(\omega)=
g\otimes\omega+\omega\otimes g- d\omega$. The two operations are mutual inverses (see \href{http://www.newton.ac.uk/programmes/NCG/seminars/080411301.pdf}{lectures} by Brzezinski or the arxiv version \href{http://arxiv.org/abs/math/0608170}{math/0608170}).

Moreover [[connection for coring|flat connections]] for a semi-free dga are in $1$-$1$ correspondence with the comodules over the corresponding coring with a group-like element.

\hypertarget{relation_to_lie_algebroids}{}\subsection*{{Relation to Lie $\infty$-algebroids}}\label{relation_to_lie_algebroids}

One can identify semifree [[differential graded algebra]]s in non-negative degree with Chevalley--Eilenberg algebras of (degreewise finite dimensional) [[Lie infinity-algebroid]]s

At least when the algebra in degree $0$ is of the form $C^\infty(X)$ for some space $X$, which then is the space of objects of the [[Lie infinity-algebroid]]. But if it is a more general algebra in degree $0$ one can think of a suitably generalized $L_\infty$-algebroid, for instance with a noncommutative space of objects. This generalizes the step from [[Lie algebroid]]s to Lie--Rinehart pairs.

\hypertarget{terminology}{}\subsection*{{Terminology}}\label{terminology}

Sometimes semi-free DGAs are called \emph{quasi-free}, but this is in collision with the terminology about formal smoothness of noncommutative algebras, i.e. quasi-free algebras in the sense of Cuntz and Quillen (and with extensions to homological smootheness of dg-algebras by Kontsevich).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Sullivan algebra]]


\item [[L-infinity algebra]], [[super L-infinity algebra]], ``[[FDA]]''



\end{itemize}
[[!redirects semifree dgas]] [[!redirects semi-free dga]] [[!redirects semi-free dgas]] [[!redirects semi-free differential graded algebra]] [[!redirects semi-free differential graded algebras]] [[!redirects semifree differential graded algebra]] [[!redirects semifree differential graded algebras]]

[[!redirects quasifree dgas]] [[!redirects quasi-free dga]] [[!redirects quasi-free dgas]] [[!redirects quasi-free differential graded algebra]] [[!redirects quasi-free differential graded algebras]] [[!redirects quasifree differential graded algebra]] [[!redirects quasifree differential graded algebras]]

[[!redirects semifree dg-algebra]] [[!redirects semifree dg-algebras]]

[[!redirects semifree dgc-algebra]] [[!redirects semifree dgc-algebras]]

[[!redirects semi-free dg-algebra]] [[!redirects semi-free dg-algebras]]

[[!redirects semi-free dgc-algebra]] [[!redirects semi-free dgc-algebras]]

[[!redirects semifree differential graded-commutative algebra]] [[!redirects semifree differential graded-commutative algebras]]



\end{document}