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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), | |
| 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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