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Graded algebra

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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).

1 Graded rings
2 Graded modules
3 Graded algebras
4 G-graded rings and algebras
5 Anticommutativity
- 5.1 Examples
6 See also
7 References

Graded rings

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

<math>A = \bigoplus_{n\in \mathbb N}A_n = A_0 \oplus A_1 \oplus A_2 \oplus \cdots</math>

such that the ring multiplication maps

<math> A_s \times A_r \rightarrow A_{s + r}.</math>

Explicitly this means that whenever

<math>x \in A_s, y \in A_r \implies xy \in A_{s+r}</math>

and so

<math> A_s A_r \subseteq A_{s + r}.</math>

Elements of <math>A_n</math> are known as homogeneous elements of degree n. An ideal or other subset <math>\mathfrak{a}</math> ⊂ A is homogeneous if for every element a ∈ <math>\mathfrak{a}</math>, the homogeneous parts of a are also contained in <math>\mathfrak{a}.</math> If I is a homogeneous ideal in A, then <math>A/I</math> is also a graded ring, and has decomposition

<math>A/I = \bigoplus_{n\in \mathbb N}(A_n + I)/I </math>.

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

Graded modules

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

<math>M = \bigoplus_{i\in \mathbb N}M_i ,</math>

and

<math>A_iM_j \subseteq M_{i+j}</math>

This idea is much used in commutative algebra, and elsewhere, to define under mild hypotheses a Hilbert function, namely the length of M_n 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).

Graded algebras

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

<math>E=\bigoplus_i E_i</math>

such that

A_iE_j ⊂ E_i+j, and
E_iE_j ⊂ E_i+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:

Polynomial rings. The homogeneous elements of degree n are exactly the homogeneous polynomials of degree n.
The tensor algebra T^•V of a vector space V. The homogeneous elements of degree n are the tensors of rank n, TⁿV.
The exterior algebra Λ^•V and symmetric algebra S^•V are also graded algebras.
The cohomology ring H^• in any cohomology theory is also graded, being the direct sum of the Hⁿ.

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.

G-graded rings and algebras

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

<math>A = \bigoplus_{i\in G}A_i </math>

such that

<math> A_i A_j \subseteq A_{i \cdot j} </math>

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 <math>\nabla:A\otimes A\rightarrow A</math>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.