- Graded algebra
In
mathematics , in particularabstract algebra , a graded algebra is analgebra over a field (orcommutative ring ) with an extra piece of structure, known as a gradation (or "grading").Graded rings
A graded ring "A" is a ring that has a
direct sum decomposition into (abelian) additive groups:A = igoplus_{nin mathbb N}A_n = A_0 oplus A_1 oplus A_2 oplus cdotssuch that the ring multiplication maps:A_s imes A_r ightarrow A_{s + r}.Explicitly this means that: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 = igoplus_{nin mathbb N}(A_n + I)/I .
Any (non-graded) ring "A" can be given a gradation by letting "A"0 = "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:M = igoplus_{iin 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 aHilbert function , namely the length of "M""n" as a function of "n". Again under mild hypotheses of finiteness, this function is a polynomial, theHilbert polynomial , for all large enough values of "n" (see alsoHilbert-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::E=igoplus_i E_isuch that
#"A"i"E"j ⊂ "E"i+j, and
#"E"i"E"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 ring s. The homogeneous elements of degree "n" are exactly the homogeneouspolynomial s of degree "n".
* Thetensor algebra "T"•"V" of avector space "V". The homogeneous elements of degree "n" are the tensors of rank "n", "T""n""V".
*Theexterior algebra Λ•"V" andsymmetric algebra "S"•"V" are also graded algebras.
* Thecohomology ring "H"• in anycohomology theory is also graded, being the direct sum of the "H""n".Graded algebras are much used in
commutative algebra andalgebraic geometry ,homological algebra andalgebraic topology . One example is the close relationship between homogeneouspolynomial s 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:A = igoplus_{iin 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,semigroup s may replacemonoid s.
*"G"-graded modules and algebras are defined in the same fashion as above.Examples:
* A group naturally grades the correspondinggroup ring ; similarly,monoid ring s are graded by the corresponding monoid.
*Asuperalgebra is another term for a Z2-graded algebra. Examples includeClifford algebra s. 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 abla:Aotimes A ightarrow Aof 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 asemiring to supply the gradation rather than a monoid. Specifically, a signed semiring consists of a pair (Γ, ε) where Γ is asemiring and ε : Γ → Z/2Z is ahomomorphism 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.
*Asupercommutative 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 generalizationReferences
*.
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