Graded algebra

An algebra $ A $ whose additive group can be represented in the form of a (weak) direct sum of groups $ A _ {i} $, $ i = 0, 1 \dots $ where $ A _ {i} A _ {j} \subseteq A _ {i+} j $ for any $ i, j $. As a result, the additive group of a graded algebra (considered as a module over the ring of integers) is a positively graded module. As an example of a graded algebra one can quote the algebra $ A = F [ x ] $ of polynomials over a field $ F $, where $ A _ {i} $ is the subspace generated by the monomials of degree $ i $( $ A _ {0} = F $). One can also define a graded algebra $ A $ in a more general manner as an algebra whose additive group can be represented as a direct sum of groups $ A _ \alpha $, where $ \alpha $ runs through a certain commutative semi-group $ G $ and $ A _ \alpha A _ \beta \subseteq A _ {\alpha + \beta } $ for any $ \alpha , \beta \in G $. The concept of a filtered algebra is closely connected with that of a graded algebra. It is in fact possible to define, in a natural manner, an ascending filtration

$$ A = \cup _ {k \geq 0 } \mathfrak A _ {k} ,\ \ \mathfrak A _ {0} \subset \mathfrak A _ {1} \subset \dots ,\ \ \mathfrak A _ {k} = \ \sum _ {i = 0 } ^ { k } A _ {i} , $$

on each graded algebra $ A = \sum _ {i \geq 0 } A _ {i} $. Conversely, if $ A = \cup _ {k \geq 0 } \mathfrak A _ {k} $ is a filtered algebra $ ( \mathfrak A _ {0} \subset \mathfrak A _ {1} \subset \dots, \mathfrak A _ {i} \mathfrak A _ {j} \subset \mathfrak A _ {i+} j ) $, then one defines the graded algebra $ \mathop{\rm gr} A = GA = \sum _ {i \geq 0 } A _ {i} $( where $ A _ {i} = \mathfrak A _ {i} / \mathfrak A _ {i-} 1 $, $ A _ {0} = \mathfrak A _ {0} $), which is called the graded algebra associated with $ A $. A graded ring is defined in a similar manner.

Comments

For an arbitrary group $ G $ one may define a gradation of type $ G $ on an algebra $ A $, i.e. $ A = \oplus _ {\sigma \in G } A _ \sigma $ where each $ A _ \sigma $ is an additive subgroup of $ A $ and $ A _ \sigma A _ \tau \subset A _ {\sigma + \tau } $ for all $ \sigma , \tau \in G $. The group algebra $ k G $ over the field $ k $, as well as crossed products $ k \star G $ defined by a group morphism $ \Phi : G \rightarrow \mathop{\rm Aut} ( k) $ and a $ 2 $- cocycle $ c \in H ^ {2} ( G, k ^ {*} ) $, are examples of $ G $- graded algebras (see also Cross product). The use of $ \mathbf Z $- gradations that are not necessarily positive gradations allows one to consider the graded rings associated to $ I $- adic filtrations on a ring $ R $; for an ideal $ I $ of $ R $ the $ I $- adic filtration is given by a descending chain $ R \supset I \supset I ^ {2} \supset \dots \supset I ^ {n} \supset \dots $, and then $ G ( R) = \oplus _ {n \in \mathbf N } I ^ {n} / I ^ {n+} 1 $, where $ G ( R) _ {-} n = I ^ {n} / I ^ {n+} 1 $ is negatively graded.

References

[a1]	N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , Hermann (1961) pp. Chapt. 3. Graduations, filtrations, et topologies
[a2]	C. Nâstâsescu, F. van Oystaeyen, "Graded ring theory" , North-Holland (1982)