Definitions

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading).

A graded ring A is a ring that has a direct sum decomposition into (abelian) additive groups

$A = bigoplus_\left\{nin mathbb N\right\}A_n = A_0 oplus A_1 oplus A_2 oplus cdots$
such that the ring multiplication maps
$A_s times A_r rightarrow A_\left\{s + r\right\}.$
Explicitly this means that
$x in A_s, y in A_r implies xy in A_\left\{s+r\right\}$
and so
$A_s A_r subseteq A_\left\{s + r\right\}.$

Elements of $A_n$ are known as homogeneous elements of degree n. An ideal or other subset $mathfrak\left\{a\right\}$A is homogeneous if for every element a$mathfrak\left\{a\right\}$, the homogeneous parts of a are also contained in $mathfrak\left\{a\right\}.$

If I is a homogeneous ideal in A, then $A/I$ is also a graded ring, and has decomposition

$A/I = bigoplus_\left\{nin mathbb N\right\}\left(A_n + I\right)/I$.

Any (non-graded) ring A can be given a gradation by letting A0 = A, and Ai = 0 for i > 0. This is called the trivial gradation on A.

The corresponding idea in module theory is that of a graded module, namely a module M over a graded ring A such that also

$M = bigoplus_\left\{iin mathbb N\right\}M_i ,$

and

$A_iM_j subseteq M_\left\{i+j\right\}$

This idea is much used in commutative algebra, and elsewhere, to define under mild hypotheses a Hilbert function, namely the length of Mn as a function of n. Again under mild hypotheses of finiteness, this function is a polynomial, the Hilbert polynomial, for all large enough values of n (see also Hilbert-Samuel polynomial).

A graded algebra over a graded ring A is an A-algebra E which is both a graded A-module and a graded ring in its own right. Thus E admits a direct sum decomposition
$E=bigoplus_i E_i$
such that

1. AiEjEi+j, and
2. EiEjEi+j.

Often when no grading on A is specified, it is assumed that A receives the trivial gradation, in which case one may still talk about graded algebras over A without risk of confusion.

Examples of graded algebras are common in mathematics:

Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra and algebraic topology. One example is the close relationship between homogeneous polynomials and projective varieties.

We can generalize the definition of a graded ring using any monoid G as an index set. A G-graded ring A is a ring with a direct sum decomposition

$A = bigoplus_\left\{iin G\right\}A_i$
such that
$A_i A_j subseteq A_\left\{i cdot j\right\}$

Remarks:

• A graded algebra is then the same thing as a N-graded algebra, where N is the monoid of non-negative integers.
• If we do not require that the ring have an identity element, semigroups may replace monoids.
• G-graded modules and algebras are defined in the same fashion as above.

Examples:

• A group naturally grades the corresponding group ring; similarly, monoid rings are graded by the corresponding monoid.
• A superalgebra is another term for a Z2-graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

In category theory, a G-graded algebra A is an object in the category of G-graded vector spaces, together with a morphism $nabla:Aotimes Arightarrow A$of the degree of the identity of G.

Anticommutativity

Some graded rings (or algebras) are endowed with an anticommutative structure. This notion requires the use of a semiring to supply the gradation rather than a monoid. Specifically, a signed semiring consists of a pair (Γ, ε) where Γ is a semiring and ε : Γ → Z/2Z is a homomorphism of additive monoids. An anticommutative Γ-graded ring is a ring A graded with respect to the additive structure on Γ such that:
xy=(-1)ε (deg x) ε (deg y)yx, for all homogeneous elements x and y.

Examples

• An exterior algebra is an example of an anticommutative algebra, graded with respect to the structure (Z≥ 0, ε) where ε is the homomorphism given by ε(even) = 0, ε(odd) = 1.
• A supercommutative algebra (sometimes called a skew-commutative associative ring) is the same thing as an anticommutative (Z/2Z, ε) -graded algebra, where ε is the identity endomorphism for the additive structure.