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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

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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

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Notes

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References

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  • 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.


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Graded structure

}I^{n}M/I^{n+1}M} . A differential graded module, differential graded Z {\displaystyle \mathbb {Z} } -module or DG-module is a graded module M {\displaystyle

Graded ring

gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is

Differential graded category

are endowed with the additional structure of a differential graded Z {\displaystyle \mathbb {Z} } -module. In detail, this means that Hom ⁡ ( A , B ) {\displaystyle

Glossary of module theory

of the module when r is the rank of the module. differential A differential graded module or dg-module is a graded module with a differential. direct

Module (mathematics)

submodules becomes stationary after finitely many steps. Graded A graded module is a module over a graded ring R = ⨁x Rx together with a direct sum decomposition

Derivation (differential algebra)

interior product acting on differential forms. Graded derivations of superalgebras (i.e., Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebras) are often called

D-module

In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of

Derived scheme

resolution of the differential graded algebra ( B ∙ , 0 ) {\displaystyle (B_{\bullet },0)} where B ∙ {\displaystyle B_{\bullet }} is the graded algebra with