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).

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[edit] Graded rings

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

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

such that the ring multiplication maps

A_s \times A_r \rightarrow A_{s + r}.

Explicitly this means that whenever

s \in A_s, r \in A_r \implies sr \in A_{s+r}

and so

A_s A_r \subseteq A_{s + r}.


Elements of An are known as homogeneous elements of degree n. An ideal, or other set in A, is homogeneous if for every element a it contains, the homogeneous parts of a are also contained in it.

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.

[edit] 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

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

and

A_iM_j \subseteq M_{i+j}

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).

[edit] 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

E=\bigoplus_i E_i

such that

  1. AiEj ⊂ Ei+j, and
  2. EiEj ⊂ Ei+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.

[edit] 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

A = \bigoplus_{i\in G}A_i

such that

A_i A_j \subseteq A_{i \cdot j}

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:A\otimes A\rightarrow Aof the degree of the identity of G.

[edit] 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.

[edit] 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.

[edit] See also

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