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) Zbl 0075.24305

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)