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\section*{differential graded algebra}

\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{abstract_definition}{Abstract definition}\dotfill \pageref*{abstract_definition} \linebreak
\noindent\hyperlink{detailed_component_definition}{Detailed component definition}\dotfill \pageref*{detailed_component_definition} \linebreak
\noindent\hyperlink{pregraded_algebras}{Pre-graded algebras}\dotfill \pageref*{pregraded_algebras} \linebreak
\noindent\hyperlink{tensor_product}{Tensor product}\dotfill \pageref*{tensor_product} \linebreak
\noindent\hyperlink{derivations}{Derivations}\dotfill \pageref*{derivations} \linebreak
\noindent\hyperlink{nb}{N.B.}\dotfill \pageref*{nb} \linebreak
\noindent\hyperlink{predgas}{Pre-DGAs}\dotfill \pageref*{predgas} \linebreak
\noindent\hyperlink{commutative_graded_algebras_cga}{Commutative graded algebras (CGA)}\dotfill \pageref*{commutative_graded_algebras_cga} \linebreak
\noindent\hyperlink{cdgas}{CDGAs}\dotfill \pageref*{cdgas} \linebreak
\noindent\hyperlink{connectivity}{$n$-connectivity}\dotfill \pageref*{connectivity} \linebreak
\noindent\hyperlink{filtrations}{Filtrations}\dotfill \pageref*{filtrations} \linebreak
\noindent\hyperlink{example_word_length_filtration}{Example: Word length filtration.}\dotfill \pageref*{example_word_length_filtration} \linebreak
\noindent\hyperlink{free_gas__the_tensor_algebra}{Free GAs: $T(V)$, the tensor algebra}\dotfill \pageref*{free_gas__the_tensor_algebra} \linebreak
\noindent\hyperlink{free_cgas_}{Free CGAs: $\bigwedge V$}\dotfill \pageref*{free_cgas_} \linebreak
\noindent\hyperlink{word_length_filtrations_on__and_}{Word length filtrations on $\bigwedge V$ and $T(V)$.}\dotfill \pageref*{word_length_filtrations_on__and_} \linebreak
\noindent\hyperlink{sum_and_product_of_cdgas}{Sum and Product of CDGAs.}\dotfill \pageref*{sum_and_product_of_cdgas} \linebreak
\noindent\hyperlink{koszul_convention}{Koszul convention}\dotfill \pageref*{koszul_convention} \linebreak
\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak
\noindent\hyperlink{cosimplicial_algebras}{Cosimplicial algebras}\dotfill \pageref*{cosimplicial_algebras} \linebreak
\noindent\hyperlink{dgcoalgebra}{dg-coalgebra}\dotfill \pageref*{dgcoalgebra} \linebreak
\noindent\hyperlink{homological_smoothness}{Homological smoothness}\dotfill \pageref*{homological_smoothness} \linebreak
\noindent\hyperlink{formal_dgalgebra}{Formal dg-algebra}\dotfill \pageref*{formal_dgalgebra} \linebreak
\noindent\hyperlink{curved_dgalgebra}{Curved dg-algebra}\dotfill \pageref*{curved_dgalgebra} \linebreak


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

\hypertarget{abstract_definition}{}\subsubsection*{{Abstract definition}}\label{abstract_definition}

A \textbf{dg-algebra}, or \textbf{[[differential graded algebra]]}, is equivalently

\begin{enumerate}%
\item An [[associative algebra]] $A$ which is in addition [[graded algebra]] and a [[differential algebra]] in a compatible way (with the [[differential]] [[derivation]] being of degree $\pm$ 1);


\item a [[monoid]] in the [[symmetric monoidal category]] of (possibly unbounded) [[chain complexes]] or [[cochain complex]]es with its standard structure of a [[monoidal category]] by the [[tensor product of chain complexes]];



\end{enumerate}
For the case of chain complexes we also speak of \textbf{chain algebras}.

For the case of cochain complexes we also speak of \textbf{cochain algebras}.

Recall

\begin{itemize}%
\item that the standard tensor product on (co)chain complexes is given by

\begin{displaymath}
(A\otimes B)_n = \sum_{i+j=n} A_i\otimes B_j,
\end{displaymath}
with the differential $d_{A\otimes B} = d_A\otimes B + A\otimes d_B$.


\item that a [[monoid]] in a [[monoidal category]] $C$ is an object $A$ in $C$ together with a morphism

\begin{displaymath}
\cdot : A \otimes A \to A
\end{displaymath}
that is unital and associative in the obvious sense.



\end{itemize}
This implies that a dg-algebra is, more concretely, a graded algebra $A$ equipped with a linear map $d : A \to A$ with the property that

\begin{itemize}%
\item $d\circ d = 0$


\item for $a \in A$ homogeneous of degree $k$, the element $d a$ is of degree $k+1$ (for monoids in cochain complexes) or of degree $k-1$ (for monoids in chain complexes)


\item for all $a,b \in A$ with $a$ homogeneous of degree k the \textbf{graded Leibniz rule} holds

$d (a \cdot b) = (d a) \cdot b + (-1)^k a \cdot (d b)$.



\end{itemize}
\hypertarget{detailed_component_definition}{}\subsubsection*{{Detailed component definition}}\label{detailed_component_definition}

\hypertarget{pregraded_algebras}{}\paragraph*{{Pre-graded algebras}}\label{pregraded_algebras}

A \emph{pre-graded algebra- (pre-ga) or $\mathbb{Z}$-graded algebra} is a pre-gvs, $A$, together with an algebra multiplication satisfying $A_p.A_q \subseteq A_{p+q}$ for any $p,q$. The relevant morphisms are pre-gvs morphisms which respect the multiplication. This gives a category $pre GA$.

An \emph{augmentation} of a pre-ga, $A$, is a homomorphism $\varepsilon : A \to k$. The \emph{augmentation ideal} of $(A,\varepsilon)$ is $ker \varepsilon$ and will also be denoted $\bar{A}$. The pair $(A,\varepsilon)$ is called an \emph{augmented pre-ga}.

A morphism $f:(A,\varepsilon)\to (A',\varepsilon')$ of augmented pre-gas is a homomorphism $f : A \to A'$ (thus of degree zero) such that $\varepsilon = \varepsilon ' f$. The resulting category will be written $pre \varepsilon GA$.

\hypertarget{tensor_product}{}\paragraph*{{Tensor product}}\label{tensor_product}

If $A$, $A'$ are two pre-gas, then $A\otimes A'$ is a pre-ga with

\begin{displaymath}
(a\otimes a')(b\otimes b') = (-1)^{|a' | |b|} a b \otimes a' b'
\end{displaymath}
for homogeneous $a,b \in A$, $a', b' \in A'$.

If $\varepsilon, \varepsilon$ are augmentations of $A$ and $A'$ respectively, then $\varepsilon\otimes \varepsilon'$ is an augmentation of $A\otimes A'$.

\hypertarget{derivations}{}\paragraph*{{Derivations}}\label{derivations}

Let $A$ be a pre-ga. An \emph{(algebra) derivation} of degree $p\in \mathbb{Z}$ is a linear map $\theta \in Hom_p(A,A)$ such that

\begin{displaymath}
\theta(ab) = \theta(a)b + (-1)^{p|a|}a\theta(b)
\end{displaymath}
for homogeneous $a,b \in A$.

A derivation $\theta$ of an augmented algebra, $(A, \varepsilon)$, is an algebra derivation which, in addition, satisfies $\varepsilon \theta = 0$.

Let $Der_p(A)$ be the vector space of derivations of degree $p$ of $A$, then $Der(A) = \bigoplus_p Der_p(A)$ is a pre-gvs.

\hypertarget{nb}{}\paragraph*{{N.B.}}\label{nb}

In the case of upper gradings, an element of $Der_p(A)$ sends $A^n$ into $A^{n-p}$.

\hypertarget{predgas}{}\paragraph*{{Pre-DGAs}}\label{predgas}

A \emph{differential} $\partial$ on an (augmented) pre-ga is a derivation of the (augmented) algebra of degree -1 such that $\partial\circ\partial = 0$.

The pair $(A,\partial)$ is called a \emph{pre-differential graded algebra} (pre-dga). If $A$ is augmented, then $(A,\partial)$ will be called an \emph{augmented pre-dga} $pre \varepsilon dga$.

If $(A,\partial)$ and $(A',\partial')$ are pre-dgas, then $(A,\partial)\otimes (A',\partial')$, with the conventions already noted, is one as well.

A morphism of pre-dgas (or pre-$\varepsilon$-dgas) is a morphism which is both of pre-gdvs and of pre-gas (with $\varepsilon$ as well if used). This gives categories $pre DGA$ and $pre \varepsilon DGA$.

\hypertarget{commutative_graded_algebras_cga}{}\paragraph*{{Commutative graded algebras (CGA)}}\label{commutative_graded_algebras_cga}

A pre-ga $A$ is said to be \emph{[[graded commutative algebra|graded commutative]]} if $ab = (-1)^{|a||b|}ba$ for each pair, $a, b$, of elements of $A$ of homogeneous degree.

Commutativity is preserved by tensor product.

We get obvious full subcategories $pre CDGA$ and $pre \varepsilon CDGA$ corresponding to the case with differentials.

\hypertarget{cdgas}{}\paragraph*{{CDGAs}}\label{cdgas}

A cdga is a \textbf{negatively} graded pre-cdga (in upper grading), $A= \bigoplus_{p\geq 0} A^p.$

There is an augmented variant, of course. These definitions give categories $CDGA$, etc.

See at \emph{[[differential graded-commutative algebra]]}.

\hypertarget{connectivity}{}\paragraph*{{$n$-connectivity}}\label{connectivity}

An $\varepsilon$cdga $(A,d)$ is \emph{$n$-connected} (resp. \emph{cohomologically $n$-connected} if $\bar{A}^p = 0$ for $p\leq n$, (resp. $\overline{H(A,d)}^p = 0$ for $p\leq n$). This gives subcategories $CDGA^n$ and $CDGA^{c n}$.

\hypertarget{filtrations}{}\paragraph*{{Filtrations}}\label{filtrations}

A filtration on a pre-ga, $A$, is a filtration on $A$, so that $F_p A \subseteq F_{p+1}A$, $F_p A.F_n A \subseteq F_{p+n}A$ (and, if $A$ is differential, also $\partial F_p A \subseteq F_p A$).

\hypertarget{example_word_length_filtration}{}\paragraph*{{Example: Word length filtration.}}\label{example_word_length_filtration}

Let $A$ be an augmented pre-ga and denote by

\begin{displaymath}
\bar{\mu}^p : \bigotimes^{p+1}\bar{A} \to \bar{A},
\end{displaymath}
the iterated multiplication. The \textbf{decreasing word length filtration}, $F^p A$ is given by:

\begin{displaymath}
F^0 A = A, \quad F^p A = Im\bar{\mu}^{(p-1)}  if p\geq 1.
\end{displaymath}
$Q(A) = \bar{A}/Im\bar{\mu}$ is the \emph{space of indecomposables} of A.

If $(A,\partial)$ is an augmented pre-dga, $F^p A$ is stable under $\partial$ and we get $Q(\partial)$ is a differential on $Q(A)$ and hence we get a functor $Q: pre \varepsilon DGA\to pre DGVS.$

\hypertarget{free_gas__the_tensor_algebra}{}\paragraph*{{Free GAs: $T(V)$, the tensor algebra}}\label{free_gas__the_tensor_algebra}

Given a pre-gvs, $V$, the tensor algebra generated by $V$ is given by $T(V) = \bigotimes_{n\geq 0}V^{\otimes n}$.

The augmentation sends $V$ to 0. $V^{\otimes n}$ is given the tensor product grading, and the multiplication is given by the tensor product.

\hypertarget{lemma_classical_freeness_of___is_a_left_adjoint}{}\paragraph*{{Lemma (classical: freeness of $T(V)$, $T$ is a left adjoint)}}\label{lemma_classical_freeness_of___is_a_left_adjoint}

If $A$ is a pre-ga and $f: V\to A$, a morphism to the underlying pre-gvs of $A$, there is a unique extension $\hat{f} :T(V)\to A$, which is a morphism of pre-gvs.

\hypertarget{free_cgas_}{}\paragraph*{{Free CGAs: $\bigwedge V$}}\label{free_cgas_}

This is the tensor product of the exterior algebra on the odd elements and the symmetric algebra on the even ones:

\begin{displaymath}
\bigwedge V = E(\bigoplus V_{2p+1})\otimes S(\bigoplus V_{2p}).
\end{displaymath}
It satisfies $\bigwedge(V \oplus W) \cong (\bigwedge V)\oplus (\bigwedge W)$.

If $A$ is a pre-cga, any morphism, $f : V\to A$, to the underlying pre-gvs of $A$, has a unique extension to a pre-cga morphism $\bar{f} :\bigwedge V \to A$.

If $(e_\alpha)_{\alpha \in I}$ is a homogeneous basis for $V$, $\bigwedge V$ and $T(V)$ may be written $\bigwedge((e_\alpha)_{\alpha \in I})$ and $T((e_\alpha)_{\alpha \in I})$ respectively.

\textbf{Note:}

\begin{itemize}%
\item $T(V)$ is a non-commutative polynomial algebra,


\item $\bigwedge V$ is a commutative polynomial algebra.



\end{itemize}
\hypertarget{word_length_filtrations_on__and_}{}\paragraph*{{Word length filtrations on $\bigwedge V$ and $T(V)$.}}\label{word_length_filtrations_on__and_}

On $\bigwedge V$ (resp. $T(V)$) write

\begin{displaymath}
\bigwedge V = \bigoplus_{k\geq 0}\bigwedge^k V,
\end{displaymath}
where $\bigwedge^k V$ is the subspace generated by all $v_1\wedge \ldots \wedge v_k$ with $v_1 \in V$. Then $F^p  \bigwedge V = \bigwedge^{\geq p} V = \bigoplus_{k\geq p}\bigwedge^k V$, resp. $T^k (V) = V^{\otimes k}$ and $F^p T(V) = T^{\geq p}(V) = \bigoplus_{k\geq p} T^k (V)$).

If $(\bigwedge V,d)$ is a pre-cdga, which is free as a pre-cga on a fixed $V$, then $d$ is the sum of derivations $d_k$ defined by the condition $d_k (V) \subseteq \bigwedge^k V$. There is an isomorphism between $V$ and $Q(\bigwedge V)$, which identifies $d_1$ with $Q(d)$. The derivation $d_1$ (resp. $d_2$) is called the \emph{linear part} (resp. \emph{quadratic part}) of $d$.

\hypertarget{sum_and_product_of_cdgas}{}\paragraph*{{Sum and Product of CDGAs.}}\label{sum_and_product_of_cdgas}

If $(A,d)$ and $(A',d')$ are two cdgas, their (categorical) \emph{sum} (i.e. coproduct) is their tensor product, $(A,d)\otimes(A',d' )$, whilst their product is the `direct sum', $(A,d)\oplus (A',d' )$.

\hypertarget{koszul_convention}{}\paragraph*{{Koszul convention}}\label{koszul_convention}

Given a permutation $\sigma$ of a graded object $(x_1, \ldots, x_p)$, the \emph{Koszul sign}, $\varepsilon(\sigma)$ is defined by

\begin{displaymath}
x_1\wedge \ldots \wedge x_p = \varepsilon(\sigma)x_{\sigma(1)} \wedge \ldots \wedge x_{\sigma(p)}
\end{displaymath}
in $\bigwedge(x_1, \ldots, x_p )$. We note that although we write $\varepsilon(\sigma)$, $\sigma$ does not suffice to define it as it depends also on the degrees of the various $x_i$.

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

Baues (in his book on \emph{Algebraic Homotopy}) has suggested using the terminology \textbf{chain algebra} for positively graded differential algebras and \textbf{cochain algebras} for the negatively graded ones. This seems to be a very useful convention.

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

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

\end{itemize}
\hypertarget{model_category_structure}{}\subsubsection*{{Model category structure}}\label{model_category_structure}

There is a standard [[model category]] structure on $dgAlg$.See [[model structure on dg-algebras]].

\hypertarget{cosimplicial_algebras}{}\subsubsection*{{Cosimplicial algebras}}\label{cosimplicial_algebras}

The [[monoidal Dold-Kan correspondence]] effectively identifies non-negatively graded [[chain complex]] algebras with simplicial algebras, and non-negatively graded [[cochain complex]] algebras with [[cosimplicial algebra]]s.

Since cosimplicial algebras have a fundamental interpretation dual to [[∞-space]], as described at [[∞-quantity]], this can be understood as explaining the great role differential graded algebras are playing in various context, suchh as notably in

\begin{itemize}%
\item [[rational homotopy theory]].

\end{itemize}
\hypertarget{dgcoalgebra}{}\subsubsection*{{dg-coalgebra}}\label{dgcoalgebra}

Dually, a [[comonoid]] in [[chain complex]]es is a [[dg-coalgebra]].

\hypertarget{homological_smoothness}{}\subsubsection*{{Homological smoothness}}\label{homological_smoothness}

A dga $A$ is \textbf{[[homologically smooth dga|homologically smooth]]} if as a dg-bimodule $_A A_A$ over itself it has a bounded resolution by finitely generated projective dg-bimodules.

\hypertarget{formal_dgalgebra}{}\subsubsection*{{Formal dg-algebra}}\label{formal_dgalgebra}

A dg-algebra $A$ is a [[formal dg-algebra]] if there exists a morphism

\begin{displaymath}
A \to H^\bullet(A)
\end{displaymath}
to its [[chain homology and cohomology|chain (co)homology]] (regarded as a dg-algebra with trivial differential) that is a [[quasi-isomorphism]].

\hypertarget{curved_dgalgebra}{}\subsubsection*{{Curved dg-algebra}}\label{curved_dgalgebra}

\begin{itemize}%
\item [[curved dg-algebra]]

\end{itemize}
[[!redirects differential graded algebras]] [[!redirects differential graded module]] [[!redirects chain algebra]] [[!redirects cochain algebra]] [[!redirects dg-algebra]] [[!redirects dg-algebras]]

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

[[!redirects commutatiev dg-algebra]] [[!redirects commutative dg-algebras]]

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



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