Graded

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). more...

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Graded rings

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

such that the ring multiplication maps

Explicitly this means that whenever

and so

Elements of A_n 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 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

and

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

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.;

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