In algebra, a differential graded module, or dg-module, is a Z {\displaystyle \mathbb {Z} } -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 N 0 {\displaystyle \mathbb {N} _{0}} -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.
See also Differential graded Lie algebra
Notes
References Iyengar, Srikanth; Buchweitz, Ragnar-Olaf; Avramov, Luchezar L. (2006-02-16). "Class and rank of differential modules". Inventiones Mathematicae. 169: 1–35. arXiv:math/0602344. doi:10.1007/s00222-007-0041-6. S2CID 16078533. Henri Cartan, Samuel Eilenberg, Homological algebra Fresse, Benoit (21 April 2017). Homotopy of Operads and Grothendieck-Teichmuller Groups. Mathematical Surveys and Monographs. Vol. 217. American Mathematical Soc. ISBN 978-1-4704-3481-6. Available online.