Graded algebra

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

The grading is a direct sum decomposition of the algebra into modules indexed by a monoid, usually the set of the non negative integers with the addition, such that the product of two elements belonging to two summands of the grading results in an element in the summand indexed by the product of the indices.

A graded ring is a graded algebra over the integers.

Graded ring

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

$A = \bigoplus_{n\in \mathbb N}A_n = A_0 \oplus A_1 \oplus A_2 \oplus \cdots$

such that the ring multiplication satisfies

$x \in A_s, y \in A_r \implies xy \in A_{s+r}$

and so

$A_s A_r \subseteq A_{s + r}.$

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

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

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

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 module

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

$M = \bigoplus_{i\in \mathbb N}M_i ,$

and

$A_iM_j \subseteq M_{i+j}.$

Graded modules may be considered over non graded rings by giving the trivial gradation to the ring. This allows to consider a sequence of modules as a single graded module. This is used in homological algebra to extend to chain complexes some module constructions like direct sum or tensor product.

Graded algebra

An algebra A over a ring R is a graded algebra if it is graded as a ring.

In the usual case where the ring R is not graded (in particular if R is a field), it is given the trivial grading (every element of "R" is of grade 0). Thus R⊆A₀ and the A_i are R modules.

In the case where the ring R is also a graded ring, then one requires that

$A_iR_j \subseteq A_{i+j}$

and

$R_iA_j \subseteq A_{i+j}$

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

Above definitions have been generalized to gradings ring using any monoid G as an index set. A G-graded ring A is a ring with a direct sum decomposition

$A = \bigoplus_{i\in G}A_i$

such that

$A_i A_j \subseteq A_{i \cdot j}.$

The notion of "graded ring" now becomes the same thing as a N-graded ring, where N is the monoid of non-negative integers under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set N with any monoid G.

Remarks:

• If we do not require that the ring have an identity element, semigroups may replace monoids.

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).

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.

See also

• Graded vector space
• Graded category
• Differential graded algebra
• Graded Lie algebra
• Filtered algebra, a generalization

References

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR1878556 .
• Bourbaki, N. (1974) Algebra I (Chapters 1-3), ISBN 978-3-540-64243-5, Chapter 3, Section 3.
• David S. Dummit & Richard M. Foote (2003). Abstract Algebra. Wiley. ISBN 978-0-471-43334-7.