# Graded module

A module $ A $
that can be represented as the direct sum of its submodules $ A _ {n} $(
the index $ n $
runs through all integers; some of the submodules $ A _ {n} $
may be trivial). A module $ A $
is called positively graded if $ A _ {n} = 0 $
for all $ n < 0 $,
and negatively graded if $ A _ {n} = 0 $
for all $ n > 0 $.
The non-zero elements of $ A _ {n} $
are called homogeneous elements of degree $ n $.
A submodule $ B $
of a graded module $ A $
is said to be homogeneous if it can be decomposed into a direct sum of submodules $ B _ {n} $
such that $ B _ {n} \subseteq A _ {n} $
for any integer $ n $;
$ B $
is then a graded module. If $ B $
is a homogeneous submodule of a graded module $ A $,
then the quotient module $ \overline{A}\; = A/B $
is also a graded module, i.e. $ \overline{A}\; = \sum \overline{A}\; _ {n} $,
where $ \overline{A}\; _ {n} $
is the image of the submodule $ A _ {n} $
under the natural homomorphism $ A \rightarrow A/B $,
$ \overline{A}\; _ {n} \simeq A _ {n} /B _ {n} $.
Graded modules are extensively used in homological algebra.

#### References

[1] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) |

#### Comments

A linear mapping between graded modules is a graded morphism if it respects the degree of homogeneous elements. The category of graded modules and graded morphisms is a Grothendieck category. Gradations by arbitrary groups may be introduced in a similar way. The gradations by the integers play an important role in the theory of projective algebraic varieties or schemes.

#### References

[a1] | C. Nâstâsescu, F. van Oystaeyen, "Graded ring theory" , North-Holland (1982) |

**How to Cite This Entry:**

Graded module.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Graded_module&oldid=47109