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| In [[mathematics]], in particular [[abstract algebra]] and [[topology]], a '''differential graded algebra''' is a [[graded algebra]] with an added [[chain complex]] structure that respects the algebra structure.
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| __TOC__
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| == Definition ==
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| A '''differential graded algebra''' (or simply '''DG-algebra''') ''A'' is a graded algebra equipped with a map <math>d\colon A \to A</math> which is either degree 1 (cochain complex convention) or degree <math>-1</math> (chain complex convention) that satisfies two conditions:
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| :(i) <math>d \circ d=0</math>
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| :This says that ''d'' gives ''A'' the structure of a [[chain complex]] or [[cochain complex]] (accordingly as the differential reduces or raises degree).
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| :(ii) <math>d(a \cdot b)=(da) \cdot b + (-1)^{|a|}a \cdot (db)</math>.{{Anchor|Graded Leibniz rule}}
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| :This says that the [[chain complex|differential]] ''d'' respects the '''graded [[Leibniz rule]]'''.
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| A ''DGA'' is an augmented DG-algebra, or differential graded augmented algebra{{clarify|reason=not clear what augmented means or whether these terms mean the same as “differential graded algebra”|post-text=: “augmented”?, synonyms for “DG-algebra”?|date=February 2014}} (the terminology
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| is due to Henri Cartan).<ref>H. Cartan, Sur les groupes d'Eilenberg-Mac Lane H(Π,n),
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| Proc. Nat. Acad. Sci. U. S. A. 40, (1954). 467–471</ref>
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| Many sources use the term ''DGAlgebra'' for a DG-algebra.
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| == Examples of DGAs ==
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| *The [[Koszul complex]] is a DGA.
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| *The [[Tensor algebra]] is a DGA with differential similar to that of the Koszul complex.
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| *The [[Singular cohomology]] with coefficients in a ring is a DGA; the differential is given by the [[Bockstein homomorphism]], and the product given by the [[cup product]].
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| *[[Differential forms]] on a [[manifold]], together with the [[Exterior derivative|exterior derivation]] and the [[Exterior algebra|wedge-product]] form a DGA.
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| == Other facts about DGAs ==
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| * The ''[[Homology (mathematics)|homology]]'' <math>H_*(A) = \ker(d) / \operatorname{im}(d)</math> of a
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| DG-algebra <math>(A,d)</math> is a graded algerba. The homology of a DGA is an augmented algebra.
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| == See also ==
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| * [[Chain complex]]
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| * [[Commutative ring spectrum]]
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| * [[Derived scheme]]
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| * [[Differential graded category]]
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| * [[Differential graded Lie algebra]]
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| * [[Graded (mathematics)]]
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| * [[Graded algebra]]
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| ==References==
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| {{reflist}}
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| * {{Citation | last1=Manin | first1=Yuri Ivanovich | author1-link=Yuri Ivanovich Manin | last2=Gelfand | first2=Sergei I. | title=Methods of Homological Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-43583-9 | year=2003}}, see chapter V.3
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| [[Category:Algebras]]
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| [[Category:Differential algebra]]
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| {{algebra-stub}}
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| [[de:Graduierung (Algebra)]]
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| [[es:Álgebra graduada]]
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| [[ru:Градуированная алгебра]]
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Alyson Meagher is the title her parents gave her but she doesn't like when individuals use her full title. Her family members life in Alaska but her husband desires them to move. What me and my family love is bungee jumping but I've been taking on new things recently. My day occupation is a journey agent.
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