differential graded module
{{short description|Mathematical concept}}
In algebra, a differential graded module, or dg-module, is a -graded module together with a differential; i.e., a square-zero graded endomorphism of the module of degree 1 or −1, depending on the convention. In other words, it is a chain complex having a structure of a module, while a differential graded algebra is a chain complex with a structure of an algebra.
In view of the module-variant of Dold–Kan correspondence, the notion of an -graded dg-module is equivalent to that of a simplicial module; "equivalent" in the categorical sense; see {{sectionlink||The Dold–Kan correspondence}} below.
The Dold–Kan correspondence
Given a commutative ring R, by definition, the category of simplicial modules are simplicial objects in the category of R-modules; denoted by sModR. Then sModR can be identified with the category of differential graded modules which vanish in negative degrees via the Dold-Kan correspondence.{{harvnb|Fresse|2017}}{{page needed|date=January 2017}}
See also
Notes
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References
- {{Cite journal|last1=Iyengar|first1=Srikanth|last2=Buchweitz|first2=Ragnar-Olaf|last3=Avramov|first3=Luchezar L.|date=2006-02-16|title=Class and rank of differential modules|journal=Inventiones Mathematicae|volume=169|pages=1–35|language=en|doi=10.1007/s00222-007-0041-6|arxiv=math/0602344|s2cid=16078533 }}
- Henri Cartan, Samuel Eilenberg, Homological algebra
- {{cite book |last1=Fresse |first1=Benoit |title=Homotopy of Operads and Grothendieck-Teichmuller Groups |series=Mathematical Surveys and Monographs |volume=217 |date=21 April 2017 |publisher=American Mathematical Soc. |isbn=978-1-4704-3481-6 |url=https://books.google.com/books?id=zQ24DgAAQBAJ |language=en}} [http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/ Available online].
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