Differential Graded Algebras
Introduction
A differential graded algebra is a graded algebra equipped with a degree $-1$ derivation whose square is zero. It is the smallest structure in which an algebra and a complex are the same object, and it is the ambient language of the homological constructions of this Part: the Koszul complex of Koszul Duality, the Hochschild complex of Deformation Quantization, the bar and cobar constructions of the operadic theory, and the resolutions that compute the derived functors of Homological Algebra are all differential graded algebras or modules over them.
The article is the ninth of the category and opens the homotopical layer of this Part. It follows Koszul Duality, whose resolution criterion is re-read here as the statement that a Koszul algebra is the homology of a cobar construction, and it precedes, and for which the differential graded language fixed here is the working language. The three structural results recorded are the homotopy theory of differential graded modules — homotopies, homotopy equivalences and the statement that a quasi-isomorphism between semi-free differential graded modules is a homotopy equivalence — the bar and cobar adjunction $\Omega \dashv B$ with the twisting morphisms that mediate it, and the formality of the Koszul and Hochschild complexes.
The article is algebraic. The homotopy category and the derived category are described algebraically, as localisations of categories of complexes at the quasi-isomorphisms; the model-category, simplicial and stable-homotopy formulations belong to Part II, where the topological constructions they need are available, and they are deferred. No manifold, no differential form on a manifold, no topology and no norm is used.
Throughout, $k$ is a commutative ring with identity, and later a field; a graded object is $\mathbb{Z}$-graded with the Koszul sign rule, so that for a graded algebra $A = \bigoplus_{n\in\mathbb{Z}}A_n$ one has $A_m\cdot A_n \subseteq A_{m+n}$. An element is homogeneous unless stated, and $\lvert a\rvert$ is the degree of a homogeneous $a$.
Graded Algebras and the Koszul Sign Rule
Definition. A graded $k$-algebra is a $k$-module $A$ graded by $\mathbb{Z}$, with $A_mA_n \subseteq A_{m+n}$, and a unit in degree $0$. It is graded-commutative if
$$ ab = (-1)^{\lvert a\rvert\lvert b\rvert}ba $$
for homogeneous $a,b$. A graded left module over $A$ is a graded $k$-module $M$ with $A_mM_n \subseteq M_{m+n}$.
Definition. The graded tensor product $M\otimes_kN$ of two graded $k$-modules is the graded module with $(M\otimes_kN)_n = \bigoplus_{i+j=n}M_i\otimes_kN_j$; for a graded algebra $A$ and graded modules $M,N$ the signed interchange is
$$ \tau : M\otimes_k N \to N\otimes_k M, \qquad \tau(m\otimes n) = (-1)^{\lvert m\rvert\lvert n\rvert}n\otimes m . $$
This sign is not a convention of convenience: it is forced by the requirement that the interchange composed with itself be the identity on the one hand and that the suspension functor be compatible with the product on the other, and it is the Koszul sign rule. With it, the tensor product of two graded algebras is a graded algebra, and a graded algebra that is graded-commutative has its opposite algebra identified with itself.
Example. The exterior algebra $\Lambda(V)$ on a finite-dimensional space, with the generators in degree $1$, is graded-commutative by the sign rule, since $vw = (-1)^{\lvert v\rvert\lvert w\rvert}wv = -wv$ for odd $v,w$. The symmetric algebra $\operatorname{Sym}(V)$ with generators of even degree is graded-commutative; a polynomial algebra with generators in odd degrees is not, and the graded-commutative version of it is the exterior algebra in those generators. The two algebras andare therefore the even and the odd incarnations of one construction, and the sign rule is what distinguishes them.
Proposition. Let $A$ be a graded algebra. Then the tensor product $A\otimes_k A^{\mathrm{op}}$ with the multiplication $(a\otimes b)(a'\otimes b') = (-1)^{\lvert b\rvert\lvert a'\rvert}aa'\otimes bb'$ is a graded algebra, the enveloping algebra of the graded setting, and graded left $A\otimes_kA^{\mathrm{op}}$-modules are the graded $A$-bimodules.
Proof. The associativity is a direct computation in which each crossing of a pair of factors contributes one sign; the signs cancel in exactly the four ways needed.
Differential Graded Algebras
Definition. A differential graded algebra (a DGA) over $k$ is a graded $k$-algebra $A$ with a $k$-linear map $d : A \to A$ of degree $-1$ satisfying
$$ d^2 = 0, \qquad d(ab) = (da)b + (-1)^{\lvert a\rvert}a(db) . $$
The map $d$ is the differential and the second identity is the graded Leibniz rule. If $A$ is graded-commutative as an algebra with this differential, it is a commutative DGA or a DG commutative algebra. The elements of $\ker d$ are the cycles and those of $\operatorname{im}d$ the boundaries; the quotient $H_\bullet(A) = \ker d/\operatorname{im}d$ is the homology of $A$, a graded algebra, and by the Leibniz rule $H_\bullet(A)$ is graded-commutative when $A$ is.
Definition. A morphism of DGAs is a degree-$0$ algebra map commuting with the differentials. A DG module over a DGA $A$ is a graded $A$-module $M$ with a differential $d_M$ of degree $-1$ satisfying
$$ d_M(am) = (da)m + (-1)^{\lvert a\rvert}a\,d_M(m) , $$
and the homology $H_\bullet(M)$ is a graded module over $H_\bullet(A)$. The category of DG $A$-modules and the category of DGAs are abelian only in the mildest cases; the correct invariants are the homologies, and the correct morphisms are those that induce isomorphisms on homology.
Definition. A morphism $f : A \to B$ of DGAs or DG modules is a quasi-isomorphism if $H_\bullet(f)$ is an isomorphism. Two DGAs are quasi-isomorphic if they are connected by a zig-zag of quasi-isomorphisms.
Example (the Koszul complex). Let $A = A(V,R)$ be a Koszul algebra with dual $A^!$, and let $K(A) = A\otimes_k(A^!)^*$ be its Koszul complex of Koszul Duality. Then $K(A)$ is a DG $A$-module whose differential has degree $-1$: the pair (differential, action of $A$) satisfies the Leibniz rule because $d$ is $A$-linear up to the sign recorded by the grading. When $A$ itself is given the structure of a DGA — for instance when $A$ is generated in degree $1$ and the Koszul dual is placed in complementary degrees — the Koszul complex is simultaneously a DGA and a resolution, and the Koszulity of $A$ says that $H_0(K(A)) = k$ and $H_n(K(A)) = 0$ for $n\neq0$.
Example (the Hochschild complex). Let $A$ be a flat $k$-algebra and $C^\bullet(A,A)$ its Hochschild complex with the cup product of Deformation Quantization,
$$ (f\smile g)(a_1,\dots,a_{m+n}) = f(a_1,\dots,a_m)g(a_{m+1},\dots,a_{m+n}) . $$
The Hochschild differential $\delta$ has degree $+1$ and the cup product is associative, so $C^\bullet(A,A)$ with the cohomological grading $C^n$ in degree $-n$ is a DGA, and the Gerstenhaber bracket of the same article is the additional structure of degree $-1$ that makes it a Gerstenhaber algebra. The cohomology $HH^\bullet(A,A)$ is its homology. The Hochschild complex is the standard noncommutative instance of a DGA and the basic example of the layer.
Example (the Chevalley–Eilenberg complex). Let $\mathrm{G}$ be a Lie algebra over $k$, with $k$ of characteristic $0$, and let $\wedge^\bullet\mathrm{G}^*$ be the exterior algebra on the dual, placed in degrees $\geq0$. The Chevalley–Eilenberg differential
$$
(d\alpha)(x_1,\dots,x_{n+1}) = \sum_{i in which $[x_i,x_j]$ occupies the $i$-th slot and $x_j$ is omitted, satisfies $d^2 = 0$ exactly because of the Jacobi identity of $\mathrm{G}$, and turns $\wedge^\bullet\mathrm{G}^*$ into a commutative DGA; the sign convention is the standard one, in which $d\alpha(x,y) = -\alpha([x,y])$ on $1$-cochains, and the alternative conventions differ by the signs of the individual terms. Its homology is the Lie algebra cohomology in the anti-symmetric category, where the construction and its interpretation are developed; it is recorded here because it is the prototype of a commutative DGA whose differential is a bracket. Example (the trivial extension). Let $A$ be a DGA, let $a \in A$ be a central cycle of even degree $\lvert a\rvert$, and let $A\langle x\rangle/(x^2)$ be the graded algebra obtained by adjoining an odd generator $x$ of degree $\lvert a\rvert+1$ with $x^2 = 0$. There is a unique differential extending $d$ with $dx = a$, and it is well defined: $d(x^2) = (dx)x - x(dx) = ax - xa = 0$ by centrality, the sign being that of the graded Leibniz rule for an odd generator. Thus every central cycle of even degree produces a new DGA that is free over the old one, and the construction is the elementary step of the semi-free resolutions of the next section. Definition. Let $A$ be a DGA and let $M,N$ be DG $A$-modules. Two morphisms $f,g : M \to N$ of degree $0$ commuting with the differentials are homotopic, written $f \simeq g$, if there is an $A$-linear map $h : M \to N$ of degree $+1$ with $$
f - g = d_Nh + h d_M .
$$ A morphism is a homotopy equivalence if it has an inverse up to homotopy. The homotopy category $K(A)$ has the DG $A$-modules as objects and the homotopy classes of morphisms as morphisms. Proposition. Homotopy is an equivalence relation compatible with composition, so $K(A)$ is a category; a homotopy equivalence is a quasi-isomorphism, and the converse fails in general. Proof. Reflexivity, symmetry and transitivity are the standard formal consequences of the existence of the homotopy $h$; the composition is compatible because the sum of two homotopies is a homotopy for the compositions. A homotopy equivalence induces an isomorphism on homology by applying $H_\bullet$ to the two identities. The converse fails in general, and the smallest counterexample over $k = \mathbb{Z}$ is the acyclic complex $$
0 \longrightarrow \mathbb{Z} \xrightarrow{\ 2\ } \mathbb{Z} \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow 0 ,
$$ which is not contractible: a contracting homotopy would in particular provide $h : \mathbb{Z}\to\mathbb{Z}$ with $2h = \mathrm{id}_{\mathbb{Z}}$, and multiplication by $2$ is not invertible. Hence the map from this complex to the zero complex is a quasi-isomorphism that is not a homotopy equivalence. Definition. A DG $A$-module $M$ is semi-free if it is the union of a chain $M^{(0)} \subseteq M^{(1)}\subseteq\cdots$ of DG submodules with $M^{(0)}$ free and every $M^{(n+1)}/M^{(n)}$ free on generators whose differentials lie in $M^{(n)}$; equivalently, $M$ is built from free modules by the trivial extensions of the preceding section, transfinitely. A DGA $A$ is semi-free if it is semi-free as a DG module over itself. Theorem (standard). Let $A$ be a DGA and let $M$ be a DG $A$-module whose homology is bounded below. Proof (outline). Statement 1 is the small-object argument applied one generator at a time: the construction of the preceding section adjoins a generator to kill a cycle or to kill a homology class, and transfinite iteration produces $P$ together with a quasi-isomorphism. Statement 2 is proved by lifting the identity along $f$: since $P$ is semi-free, a map $P\to Q$ can be constructed by induction over the filtration, and the resulting map is homotopic to the inverse of $f$ because the difference has zero homology and $P$ is semi-free. Statement 3 is the same lifting induction, and the uniqueness up to homotopy is the statement that any two lifts differ by a homotopy constructed in the same way. The argument is the algebraic shadow of the small-object and lifting arguments of homotopy theory; the model-categorical formulation belongs to Part II. Definition. The derived category $D(A)$ of a DGA $A$ is the localisation of the homotopy category $K(A)$ at the quasi-isomorphisms: the category obtained by formally adjoining inverses to all quasi-isomorphisms. Its existence and its identification with the homotopy category of semi-free modules are proved by the standard methods of homological algebra, and the model structure that produces it belongs to Part II. Corollary. The derived category $D(A)$ is equivalent to the homotopy category of semi-free DG $A$-modules, and the derived functors $\operatorname{RHom}$ and $\otimes^{\mathbb{L}}$ are computed by semi-free resolutions and by their duals. Definition. Let $A$ be an augmented DGA over a field $k$ with augmentation $\varepsilon : A \to k$ and augmentation ideal $\bar A = \ker\varepsilon$. The bar construction $B(A)$ is the graded coalgebra $$
B(A) = \bigoplus_{n\geq0}\bar A^{\otimes n}, \qquad \bar A^{\otimes0} = k ,
$$ with the deconcatenation coproduct $\Delta(a_1\otimes\cdots\otimes a_n) = \sum_{i=0}^{n}(a_1\otimes\cdots\otimes a_i)\otimes(a_{i+1}\otimes\cdots\otimes a_n)$ and the differential $$
d(a_1\otimes\cdots\otimes a_n) = \sum_{i=1}^{n}(-1)^{\lvert a_1\rvert+\cdots+\lvert a_{i-1}\rvert}a_1\otimes\cdots\otimes da_i\otimes\cdots\otimes a_n + \sum_{i=1}^{n-1}(-1)^{\lvert a_1\rvert+\cdots+\lvert a_i\rvert}a_1\otimes\cdots\otimes a_ia_{i+1}\otimes\cdots\otimes a_n .
$$ Definition. Let $C$ be a coaugmented graded coalgebra with coaugmentation ideal $\bar C$. The cobar construction $\Omega(C)$ is the graded algebra $$
\Omega(C) = T_k(\bar C[-1]) ,
$$ the tensor algebra on the desuspension of $\bar C$, with the unique derivation extending $d(\xi) = -d_C(\xi)$ for $\xi\in\bar C$ on the generators; the sign is chosen so that the deconcatenation of the coalgebra and the concatenation of the algebra are adjoint. Theorem (bar–cobar adjunction, standard). The bar and cobar constructions form an adjoint pair $$
\Omega : \{\text{coaugmented DG coalgebras}\} \rightleftarrows \{\text{augmented DGAs}\} : B ,
$$ with $\Omega \dashv B$. The unit and counit are not isomorphisms, and the correct statement of the relation is given by twisting morphisms: for a DGA $A$ and a DG coalgebra $C$, the twisting morphisms are the degree $-1$ linear maps $\tau : C \to A$ satisfying the Maurer–Cartan equation $$
d\tau + \tau\star\tau = 0
$$ for the convolution product $\star$ on $\operatorname{Hom}_k(C,A)$, and they are in natural bijection with the DGA morphisms $\Omega(C)\to A$ and with the DG coalgebra morphisms $C \to B(A)$. Proof (outline). A map $\Omega(C)\to A$ is determined by its restriction to the generators $\bar C$, that is, by a degree $-1$ map $\tau : C\to A$; the requirement that the map be a morphism of DGAs is exactly $d\tau + \tau\star\tau = 0$, since the differential on the tensor algebra is the concatenation and the differentials. The bijection with coalgebra maps $C\to B(A)$ is the identical computation read in the opposite category. Theorem (Koszul duality via the cobar construction, standard). Let $A$ be a Koszul algebra over a field, with Koszul dual $A^!$ regarded as a graded coalgebra by dualising the multiplication. Then there is a quasi-isomorphism of DGAs $$
\Omega(A^!)\ \xrightarrow{\ \simeq\ }\ A ,
$$ so that $A$ is recovered as the homology of the cobar construction on its Koszul dual, and dually $B(A)$ is quasi-isomorphic to $A^!$. The Koszul complex of Koszul Duality is the linearisation of this statement: the bar construction is the noncommutative Koszul complex, and the cobar construction is its dual. This is the precise sense in which the two constructions of the previous article — the resolution criterion and the derived equivalence — are the same fact, and it is the reason that Koszul duality is often stated as an equivalence between the homotopy theory of algebras and the homotopy theory of coalgebras. Definition. A DGA $A$ is formal if it is quasi-isomorphic to its homology $H_\bullet(A)$ equipped with the zero differential; a DG operad is formal if it is quasi-isomorphic to its homology operad. Formality is the statement that the DGA contains no more information than its homology, and the failure of formality is measured by Massey products, the higher operations $\langle a_1,\dots,a_n\rangle$ defined when all consecutive products vanish and not defined otherwise. Proposition. Let $A$ be a Koszul algebra over a field. Then the Koszul complex computing $\operatorname{Ext}^\bullet_A(k,k)$ is formal, and $\operatorname{Ext}^\bullet_A(k,k) \cong A^!$ as a graded algebra; more generally a DGA whose homology is concentrated in degrees of a single parity, in the appropriate sense, is formal because the higher operations must land in degrees that are not represented. Proof. For a Koszul algebra the Koszul complex is a semi-free resolution of $k$ with generators in bidegrees $(i,i)$, so the comparison of the complex $\operatorname{End}_A(K(A))$ with its homology $A^!$ is a quasi-isomorphism of DGAs by the dimension count in each bidegree; no room remains for a nontrivial Massey product, and the formality follows. Theorem (Kontsevich formality; statement of the DGA form). Let $A = k[x_1,\dots,x_n]$ and let $C^\bullet(A,A)$ be its Hochschild complex as a DGA; let $T_{\mathrm{poly}}(A)$ be the graded algebra of polynomial multivectors with the zero differential and the wedge product. Then there is a quasi-isomorphism of DGAs (indeed of $L_\infty$-algebras, as in Deformation Quantization and) $$
\bigl(T_{\mathrm{poly}}(A),\ 0\bigr) \longrightarrow \bigl(C^\bullet(A,A),\ \delta\bigr)
$$ from the multivectors with zero differential to the Hochschild complex, and this quasi-isomorphism is the source of the star products of Deformation Quantization. The formality of the little disks operad, stated in Deformation Quantization, is the operadic form of the statement; the topological and model-categorical versions belong to Part II. A graded algebra is a $\mathbb{Z}$-graded $k$-module with $A_mA_n\subseteq A_{m+n}$ and the Koszul sign rule $ab = (-1)^{\lvert a\rvert\lvert b\rvert}ba$ for the graded-commutative case; the signed interchange $\tau(m\otimes n) = (-1)^{\lvert m\rvert\lvert n\rvert}n\otimes m$ makes the tensor product of graded algebras an algebra and identifies the opposite of a graded-commutative algebra with itself. A differential graded algebra is a graded algebra with a degree $-1$ map $d$ satisfying $d^2 = 0$ and the graded Leibniz rule $d(ab) = (da)b + (-1)^{\lvert a\rvert}a(db)$; its homology $H_\bullet(A)$ is a graded algebra, a DG module is a graded module with a compatible differential, and a quasi-isomorphism is a morphism inducing an isomorphism on homology. The Koszul complex of a Koszul algebra, the Hochschild complex with the cup product, the Chevalley–Eilenberg complex of a Lie algebra and the trivial extension $A\langle x\rangle/(x^2)$ with $dx = a$ for a central cycle $a$ are the basic examples. Homotopy of DG module morphisms is defined by $f-g = d_Nh + hd_M$; the homotopy category $K(A)$ is the quotient by homotopy, a homotopy equivalence is a quasi-isomorphism, and the converse fails in general. Every DG module with homology bounded below admits a semi-free resolution, quasi-isomorphisms between semi-free modules are homotopy equivalences, and the derived category $D(A)$ is the localisation of $K(A)$ at the quasi-isomorphisms, equivalent to the homotopy category of semi-free modules; the model structure producing the localisation belongs to Part II. The bar construction $B(A)$ is the tensor coalgebra on $\bar A[-1]$ with the deconcatenation coproduct and a differential combining $d$ and the product, the cobar construction $\Omega(C) = T(\bar C[-1])$ is its adjoint, and the pair is mediated by the twisting morphisms $\tau$ satisfying the Maurer–Cartan equation $d\tau + \tau\star\tau = 0$; for a Koszul algebra there is a quasi-isomorphism $\Omega(A^!) \xrightarrow{\simeq} A$, which is Koszul duality read in the language of this article. A DGA is formal when it is quasi-isomorphic to its homology with zero differential; the Koszul complex is formal, and the Kontsevich formality theorem gives a quasi-isomorphism from the multivectors with zero differential to the Hochschild complex, the source of deformation quantisation.Modules, Homotopies and Semi-free Resolutions
The Bar and Cobar Constructions
Formality
Summary
Summary of Notation
Symbol
Meaning
$k$, $A$, $M$, $N$
ground ring, graded algebra or DGA, modules
$A = \bigoplus_{n\in\mathbb{Z}}A_n$
graded algebra
$\lvert a\rvert$
degree of a homogeneous element
$ab = (-1)^{\lvert a\rvert\lvert b\rvert}ba$
graded commutativity (Koszul sign rule)
$d$, $d^2 = 0$
differential of degree $-1$
$d(ab) = (da)b + (-1)^{\lvert a\rvert}a(db)$
graded Leibniz rule
$H_\bullet(A)$, $H_\bullet(M)$
homology of a DGA, of a DG module
$\tau$
signed interchange $M\otimes_kN\to N\otimes_kM$
$A\otimes_kA^{\mathrm{op}}$
graded enveloping algebra
$K(A)$, $D(A)$
homotopy category, derived category of DG $A$-modules
$\simeq$
homotopy equivalence (on morphisms, homotopy)
$B(A)$, $\Omega(C)$
bar construction, cobar construction
$\bar A$, $\bar C$
augmentation ideal, coaugmentation ideal
$\tau : C\to A$, $d\tau + \tau\star\tau = 0$
twisting morphism, Maurer–Cartan equation
$\langle a_1,\dots,a_n\rangle$
Massey product
$T_{\mathrm{poly}}(A)$
multivectors of a commutative algebra, with the wedge product
Further Reading