Differential graded module

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 #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 modules over R; denoted by sModR. Then the category can be identified with the category of differential graded modules.[1]

gollark: Fun times, that was. We were using JS to implement that weird stacky language.
gollark: Oh, like our interpreter race.
gollark: Probably just lack of experience and/or poor teaching, but still.
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See also

References
  1. Benoit Fresse, somewhere
  • 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.
  • H. Cartan, S. Eilenberg, Homological algebra
  • Benoit Fresse, Homotopy of Operads & Grothendieck-Teichmüller Groups


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