Differential graded module
In algebra, a differential graded module, or dg-module, is a [math]\displaystyle{ \mathbb{Z} }[/math]-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 [math]\displaystyle{ \mathbb{N}_0 }[/math]-graded dg-module is equivalent to that of a simplicial module; "equivalent" in the categorical sense; see § 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.[1]
See also
References
- Iyengar, Srikanth; Buchweitz, Ragnar-Olaf; Avramov, Luchezar L. (2006-02-16). "Class and rank of differential modules" (in en). Inventiones Mathematicae 169: 1–35. doi:10.1007/s00222-007-0041-6.
- Henri Cartan, Samuel Eilenberg, Homological algebra
- Fresse, Benoit (21 April 2017) (in en). Homotopy of Operads and Grothendieck-Teichmuller Groups. Mathematical Surveys and Monographs. 217. American Mathematical Soc.. ISBN 978-1-4704-3481-6. https://books.google.com/books?id=zQ24DgAAQBAJ. Available online.
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