View source for Graded module

<!--
g0446701.png
$#A+1 = 25 n = 0
$#C+1 = 25 : ~/encyclopedia/old_files/data/G044/G.0404670 Graded module
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra", Princeton Univ. Press (1956) {{ZBL|0075.24305}}</TD></TR>
</table>

====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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Nâstâsescu,   F. van Oystaeyen,   "Graded ring theory" , North-Holland  (1982)</TD></TR></table>