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{{Short description|Algebraic structure in homological algebra}}
{{about|homological algebra|the algebraic study of differential equations|Differential algebra}}
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{{context|date=March 2023}}
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In [[mathematics]] – particularly in [[homological algebra]], [[algebraic topology]], and [[algebraic geometry]] – a '''differential graded algebra''' (or '''DGA''', or '''DG algebra''') is an [[algebra|algebraic structure]] often used to capture information about a [[topological space|topological]] or [[geometry|geometric]] space. Explicitly, a differential graded algebra is a [[graded algebra|graded]] [[associative algebra]] with a [[chain complex]] structure that is compatible with the [[algebra over a field|algebra]] structure. 

In geometry, the [[de rham cohomology|de Rham algebra]] of [[differential forms]] on a [[manifold]] has the structure of a differential graded algebra, and it encodes the [[de Rham cohomology]] of the manifold. In algebraic topology, the [[singular cohomology|singular cochains]] of a topological space form a DGA encoding the [[Cohomology#Singular_cohomology|singular cohomology]]. Moreover, American mathematician [[Dennis Sullivan]] developed a DGA to encode the [[rational homotopy type]] of topological spaces.{{sfn|Sullivan|1977}}

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== Definitions ==

Let <math>A_\bullet = \bigoplus\nolimits_{i \in \mathbb{Z}} A_i</math> be a <math>\mathbb{Z}</math>-[[graded algebra]], with product <math>\cdot</math>, equipped with a map <math>d\colon A_\bullet \to A_\bullet</math> of [[Graded_vector_space#Homomorphisms|degree]] <math>-1</math> (homologically graded) or degree <math>+1</math> (cohomologically graded). We say that <math>(A_\bullet,d,\cdot)</math> is a '''differential graded algebra''' if <math>d</math> is a differential, giving <math>A_\bullet</math> the structure of a [[chain complex]] or [[cochain complex]] (depending on the degree), and satisfies a graded [[Product rule|Leibniz rule]]. In what follows, we will denote the "degree" of a [[Graded_vector_space#Integer_gradation|homogeneous element]] <math>a\in A_i</math> by <math>|a| = i</math>. Explicitly, the map <math>d</math> satisfies the conditions{{sfn|Loday|Vallette|2012|p=28-29}}
{{ordered list|type=lower-roman
  | <math>d \circ d = 0</math>, often written <math>d^2=0</math>.
  | <math>d(a \cdot b) = (da) \cdot b + (-1)^{|a|}a \cdot (db)</math>.}}
Often one omits the differential and multiplication and simply writes <math>A_\bullet</math> or <math>A</math> to refer to the DGA <math>(A_\bullet,d,\cdot)</math>.

A [[linear map]] <math>f: A_\bullet \to B_\bullet</math> between [[graded vector space|graded vector spaces]] is said to be of '''degree ''n''''' if <math>f(A_i) \subseteq B_{i+n}</math> for all <math>i</math>. When considering (co)chain complexes, we restrict our attention to [[Chain_complex#Chain_maps|chain maps]], that is, maps of degree 0 that commute with the differentials <math>f \circ d_A = d_B \circ f</math>. The morphisms in the [[category (mathematics)|category]] of DGAs are chain maps that are also [[Algebra_over_a_field#Algebra_homomorphisms|algebra homomorphisms]].

=== Categorical Definition ===

One can also define DGAs more abstractly using [[category theory]]. There is a [[Chain_complex#Category_of_chain_complexes|category of chain complexes]] over a [[ring (mathematics)|ring]] <math>R</math>, often denoted <math>\operatorname{Ch}_R</math>, whose objects are chain complexes and whose morphisms are chain maps. We define the [[tensor product]] of chain complexes <math>(V,d_V)</math> and <math>(W,d_W)</math> by{{sfn|Weibel|1994|p=58}}
:<math> (V \otimes W)_n = \bigoplus_{i+j=n} V_i \otimes_R W_j </math>
with differential
:<math> d (v \otimes w) = (d_V v) \otimes w - (-1)^{|v|} v \otimes (d_W w) </math>
This operation makes <math>\operatorname{Ch}_R</math> into a [[symmetric monoidal category]]. Then, we can equivalently define a differential graded algebra as a [[monoid object]] in <math>\operatorname{Ch}_R</math>. Heuristically, it is an object in <math>\operatorname{Ch}_R</math> with an associative and unital multiplication.

=== Homology and Cohomology ===

Associated to any chain complex <math>(A_\bullet,d)</math> is its [[homology (mathematics)|homology]]. Since <math>d \circ d = 0</math>, it follows that <math>\operatorname{im}(d:A_{i+1} \to A_i)</math> is a subobject of <math>\operatorname{ker}(d:A_i \to A_{i-1})</math>. Thus, we can form the quotient
:<math>H_i(A_\bullet) = \operatorname{ker}(d:A_i \to A_{i-1}) / \operatorname{im}(d:A_{i+1} \to A_i)</math>
This is called the <math>i</math>th homology group, and all together they form a graded vector space <math>H_\bullet(A)</math>. In fact, the homology groups form a DGA with zero differential. Analogously, one can define the [[cohomology|cohomology groups]] of a cochain complex, which also form a graded algebra with zero differential.

Every chain map <math>f: (A_\bullet,d_A) \to (B_\bullet,d_B)</math> of complexes induces a map on (co)homology, often denoted <math>f_*: H_\bullet(A) \to H_\bullet(B)</math> (respectively <math>f^*: H^\bullet(B) \to H^\bullet(A)</math>). If this induced map is an [[isomorphism]] on all (co)homology groups, the map <math>f</math> is called a [[quasi-isomorphism]]. In many contexts, this is the natural notion of equivalence one uses for (co)chain complexes. We say a morphism of DGAs is a quasi-isomorphism if the chain map on the underlying (co)chain complexes is.

== Properties of DGAs ==

=== Commutative Differential Graded Algebras ===

A '''commutative differential graded algebra''' (or '''CDGA''') is a differential graded algebra, <math>(A_\bullet, d, \cdot)</math>, which satisfies a graded version of commutativity. Namely,
: <math>a \cdot b = (-1)^{|a| |b|} b \cdot a</math>
for homogeneous elements <math>a \in A_i, b \in A_j</math>. Many of the DGAs commonly encountered in math happen to be CDGAs, like the de Rham algebra of differential forms.

=== Differential graded Lie algebras ===

A '''[[differential graded Lie algebra]]''' (or '''DGLA''') is a differential graded analogue of a [[Lie algebra]]. That is, it is a differential graded vector space, <math>(L_\bullet, d)</math>, together with an operation <math>[,]: L_i \otimes L_j \to L_{i+j}</math>, satisfying the following graded analogues of the Lie algebra axioms.
{{ordered list|type=lower-roman
  | Graded skew-symmetry: <math>[x,y] = -(-1)^{|x| |y|} [y,x]</math> for homogeneous elements <math>x\in L_i, y \in L_j</math>.
  | Graded [[Jacobi identity]]: <math>(-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0</math>.
  | Graded [[Leibniz rule]]: <math>d [x,y] = [d x,y] + (-1)^{|x|}[x, d y] </math>.}}

An example of a DGLA is the de Rham algebra <math>\Omega^\bullet(M)</math> tensored with a [[Lie algebra]] <math>\mathfrak{g}</math>, with the bracket given by the [[Differential_form#Exterior_product|exterior product]] of the [[differential forms]] and Lie bracket; elements of this DGLA are known as [[Lie algebra–valued differential form]]s. DGLAs also arise frequently in the study of [[deformation theory|deformations]] of algebraic structures where, over a field of characteristic 0, "nice" deformation problems are described by the space of [[Maurer-Cartan form|Maurer-Cartan elements]] of some suitable DGLA.{{sfn|Kontsevich|Soibelman|p=14}}

=== Formal DGAs ===

A (co)chain complex <math>C_\bullet</math> is called '''formal''' if there is a chain map to its (co)homology <math>H_\bullet(C_\bullet)</math> (respectively <math>H^\bullet(C_\bullet)</math>), thought of as a complex with 0 differential, that is a quasi-isomorphism. We say that a DGA <math>A</math> is '''formal''' if there exists a morphism of DGAs <math>A \to H_\bullet(A)</math> (respectively <math>A \to H^\bullet(A)</math>) that is a quasi-isomorphism.{{sfn|Sullivan|1977|p=315}} This notion is important, for instance, when one wants to consider quasi-isomorphic chain complexes or DGAs as being equivalent, as in the [[derived category]].

== Examples ==

=== Trivial DGAs ===

Notice that any [[graded ring|graded algebra]] <math>A=\bigoplus\nolimits_i A_i</math> has the structure of a DGA with trivial differential, i.e., <math>d=0</math>. In particular, as noted above, the (co)homology of any DGA forms a trivial DGA, since it is a graded algebra.

=== The de-Rham algebra ===

Let <math>M</math> be a [[manifold]]. Then, the [[differential forms]] on <math>M</math>, denoted by <math>\Omega^\bullet(M)</math>, naturally have the structure of a (cohomologically graded) DGA. The graded vector space is <math>\Omega^\bullet(M)</math>, where the grading is given by form degree. This vector space has a product, given by the [[Differential_form#Exterior_product|exterior product]], which makes it into a graded algebra. Finally, the exterior derivative <math>d: \Omega^i(M) \to \Omega^{i+1}(M)</math> satisfies <math>d^2=0</math> and the graded Leibniz rule. In fact, the exterior product is graded-commutative, which makes the de Rham algebra an example of a CDGA.{{sfn|Griffiths|Morgan|2013|p=95}}

=== Singular Cochains ===

Let <math>X</math> be a [[topological space]]. Recall that we can associate to <math>X</math> its complex of [[Cohomology#Singular_Cohomology|singular cochains]] with coefficients in a [[ring]] <math>R</math>, denoted <math>(C^\bullet(X;R),d)</math>, whose cohomology is the [[Cohomology#Singular_Cohomology|singular cohomology]] of <math>X</math>. On <math>C^\bullet(X;R)</math>, one can define the [[cup product]] of cochains, which gives this cochain complex the structure of a DGA.{{sfn|Griffiths|Morgan|2013|p=96}} In the case where <math>X</math> is a smooth manifold and <math>R=\mathbb{R}</math>, the [[de Rham theorem]] states that the singular cohomology is isomorphic to the [[de Rham cohomology]] and, moreover, the cup product and exterior product of differential forms induce the same operation on cohomology.

Note, however, that while the cup product induces a graded-commutative operation on cohomology, it is not graded commutative directly on cochains. This is an important distinction, and the failure of a DGA to be commutative is referred to as the "commutative cochain problem". This problem is important because if, for any topological space <math>X</math>, one can associate a '''commutative''' DGA whose cohomology is the singular cohomology of <math>X</math> over <math>R</math>, then this CDGA determines the <math>R</math>-homotopy type of <math>X</math>.{{sfn|Griffiths|Morgan|2013|p=96}}

=== The Free DGA ===

Let <math>V</math> be a (non-graded) vector space over a [[field (mathematics)|field]] <math>k</math>. The [[tensor algebra]] <math>T(V)</math> is defined to be the graded algebra 
:<math>T(V) = \bigoplus_{i\geq 0} T^i(V) = \bigoplus_{i \geq 0} V^{\otimes i}</math>
where, by convention, we take <math>T^0(V) = k</math>. This vector space can be made into a graded algebra with the  multiplication <math>T^i(V) \otimes T^j(V) \to T^{i+j}(V)</math> given by the tensor product <math>\otimes</math>. This is the [[free algebra]] on <math>V</math>, and can be thought of as the algebra of all non-commuting polynomials in the elements of <math>V</math>.

One can give the tensor algebra the structure of a DGA as follows. Let <math>f: V \to k</math> be any linear map. Then, this extends uniquely to a [[Derivation (differential algebra)|derivation]] of <math>T(V)</math> of degree <math>-1</math> (homologically graded) by the formula{{sfn|Loday|Vallette|2012|p=5}}
:<math>d_f(v_1 \otimes \cdots \otimes v_n) = \sum_{i=1}^n (-1)^{i-1} v_1 \otimes \cdots \otimes f(v_i) \otimes \cdots \otimes v_n</math>
One can think of the minus signs on the right-hand side as coming from "jumping" the map <math>f</math> over the elements <math>v_1, \ldots, v_{i-1}</math>, which are all of degree 1 in <math>T(V)</math>. This is commonly referred to as the [[Koszul complex|Koszul sign rule]].

One can extend this construction to differential graded vector spaces. Let <math>(V_\bullet,d_V)</math> be a differential graded vector space, i.e., <math>d_V: V_i \to V_{i-1}</math> and <math>d^2=0</math>. Here we work with a homologically graded DG vector space, but this construction works equally well for a cohomologically graded one. Then, we can endow the tensor algebra <math>T(V)</math> with a DGA structure which extends the DG structure on V. The differential is given by
:<math> d(v_1 \otimes \cdots \otimes v_n) = \sum_{i=1}^n (-1)^{|v_1|+\ldots+|v_{i-1}|} v_1 \otimes \cdots \otimes d_V(v_i) \otimes \cdots \otimes v_n </math>
This is similar to the previous case, except that now the elements of <math>V</math> can have different degrees, and <math>T(V)</math> is no longer graded by the number of tensor products but instead by the sum of the degrees of the elements of <math>V</math>, i.e., <math>|v_1 \otimes \cdots \otimes v_n| = |v_1| + \ldots + |v_n|</math>.{{sfn|Loday|Vallette|2012|p=23}}

=== The Free CDGA ===

Similar to the previous case, one can also construct the free CDGA. Given a graded vector space <math>V_\bullet</math>, we define the free graded commutative algebra on it by
: <math>S(V) = \operatorname{Sym}\left( \bigoplus_{i=2k} V_i \right) \otimes \bigwedge \left( \bigoplus_{i=2k+1} V_i \right) </math>
where <math>\operatorname{Sym}</math> denotes the [[symmetric algebra]] and <math>\bigwedge</math> denotes the [[exterior algebra]]. If we begin with a DG vector space <math>(V_\bullet, d)</math> (either homologically or cohomologically graded), then we can extend <math>d</math> to <math>S(V)</math> such that <math>(S(V),d)</math> is a CDGA in a unique way.{{sfn|Felix|Oprea|Tanre|2008|p=58}}

== Models for DGAs ==

As mentioned previously, oftentimes one is most interested in the (co)homology of a DGA. As such, the specific (co)chain complex we use is less important, as long as it has the right (co)homology. Given a DGA <math>A</math>, we say that another DGA <math>M</math> is a '''model''' for <math>A</math> if it comes with a [[surjective]] DGA morphism <math>p : M \to A</math> that is a quasi-isomorphism.{{sfn|Loday|Vallette|2012|p=29}}

=== Minimal Models ===

Since one could form arbitrarily large (co)chain complexes with the same cohomology, it is useful to consider the "smallest" possible model of a DGA. We say that a DGA <math>(A,d,\cdot)</math> is a '''minimal''' if it satisfies the following conditions.{{sfn|Griffiths|Morgan|2013|p=97}}
{{ordered list|type=lower-roman
  | It is free as a graded algebra, i.e., <math>A \cong T(V)</math> for some graded vector space <math>V</math>.
  | The differential satisfies <math>d(A) \subseteq A^+ \cdot A^+</math>, where <math>A^+ = \bigoplus\nolimits_{i>0} A_i</math> consists of the positive degree parts of <math>A</math>.}}
Note that some conventions, often used in algebraic topology, additionally require that <math>A</math> be '''simply connected''', which means that <math>A^0=k</math> and <math>A^1=0</math>.{{sfn|Griffiths|Morgan|2013|p=97}} This condition on the 0th and 1st degree components of <math>A</math> mirror the (co)homology groups of a [[simply connected space]]. 

Finally, we say that <math>M</math> is a '''minimal model''' for <math>A</math> if it is both minimal and a model for <math>A</math>. The fundamental theorem of minimal models{{sfn|Loday|Vallette|2012|p=29}} states that if <math>A</math> is simply connected then it admits a minimal model, and that if a minimal model exists it is unique up to (non-unique) isomorphism.{{sfn|Deligne|Griffiths|Morgan|Sullivan|1975|p=251}}

=== The Sullivan minimal model ===
{{main|Rational homotopy theory}}

Minimal models were used with great success by Dennis Sullivan in his work on [[rational homotopy theory]]. Given a [[simplicial complex]] <math>X</math>, one can define a rational analogue of the (real) de Rham algebra: the DGA <math>A_{PL}(X)</math> of "piecewise polynomial" differential forms with <math>\mathbb{Q}</math>-coefficients. Then, <math>A_{PL}(X)</math> has the structure of a CDGA over the field <math>\mathbb{Q}</math>, and in fact the cohomology is isomorphic to the singular cohomology of <math>X</math>.{{sfn|Deligne|Griffiths|Morgan|Sullivan|1975|p=255}} In particular, if <math>X</math> is a simply connected topological space then <math>A_{PL}(X)</math> is simply connected as a DGA, thus there exists a minimal model. 

Moreover, since <math>A_{PL}(X)</math> is a CDGA whose cohomology is the singular cohomology of <math>X</math> with <math>\mathbb{Q}</math>-coefficients, it is a solution to the commutative cochain problem. Thus, if <math>X</math> is a simply connected [[CW complex]] with finite dimensional rational homology groups, the minimal model of the CDGA <math>A_{PL}(X)</math> captures entirely the rational homotopy type of <math>X</math>.{{sfn|Griffiths|Morgan|2013|p=2}}

== See also ==
* [[Differential graded Lie algebra]]
* [[Rational homotopy theory]]
* [[Homotopy associative algebra]]

==Notes==
{{reflist|colwidth=30em}}

==References==
{{refbegin}}

* {{Cite journal | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Griffiths | first2=Phillip | author2-link=Phillip Griffiths | last3=Morgan | first3=John | author3-link=John Morgan (mathematician) | last4=Sullivan | first4=Dennis | author4-link=Dennis Sullivan | title=Real homotopy theory of Kähler manifolds | year=1975 | journal=Inventiones mathematicae | volume=29 | pages=245–274 | publisher=[[Springer Publishing|Springer]] | doi=10.1007/BF01389853 }}

* {{Citation | last1=Felix | first1=Yves | last2=Oprea | first2=John | last3=Tanre | first3=Daniel | title=Algebraic Models in Geometry | year=2008 | series=Oxford Graduate Texts in Mathematics | volume=17 | publisher=[[Oxford University Press]] | location=New York | isbn=978-0-19-920651-3}}

* {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Morgan | first2=John | author2-link=John Morgan (mathematician) | title=Rational Homotopy Theory and Differential Forms | series=Progress in Mathematics | volume=16 | edition=3rd | publisher=[[Birkhäuser]] | location=New York, Heidelberg, Dordrecht, London | isbn=978-1-4614-8467-7 | year=2013}}

* {{Citation | last1=Kontsevich | first1=Maxim | author1-link=Maxim Kontsevich | last2=Soibelman | first2=Yan | author2-link=Yan Soibelman | title=Deformation Theory. I}}

* {{Citation | last1=Loday | first1=Jean-Louis | author1-link=Jean-Louis Loday | last2=Vallette | first2=Bruno | title=Algebraic Operads | series=Grundlehren der mathematischen Wissenschaften | volume=346 | publisher=[[Springer Publishing|Springer]] | location=Heidelberg, New York, Dordrecht, London | isbn=978-3-642-30362-3 | year=2012}}

* {{Citation | author1-last=Sullivan | author1-first=Dennis | author1-link=Dennis Sullivan | title=Infinitesimal computations in topology | year=1977 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=47 | pages=269–331
 | url=http://www.numdam.org/item?id=PMIHES_1977__47__269_0 | mr=0646078 | doi=10.1007/bf02684341 | hdl=10338.dmlcz/128041}}

* {{Citation | last1=Weibel | first1=Charles | author1-link=Charles Weibel | title=An Introduction to Homological Algebra | series=Cambridge Studies in Advanced Mathematics | volume=38 | publisher=[[Cambridge University Press]] | location=Cambridge | isbn=0-521-43500-5 | year=1994}}

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