Differential graded module

In algebra, a differential graded module, or dg-module, is a $\mathbb$- 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 $\mathbb_0$-graded dg-module is equivalent to that of a simplicial module; "equivalent" in the categorical sense; see 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 ''s''Mod_''R''. Then ''s''Mod_''R'' 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

References

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*Henri Cartan, Samuel Eilenberg, ''Homological algebra''
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Available online

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Algebra