Graded ring

Inmathematics,in particularabstract algebra,agraded ringis aringsuch that the underlyingadditive groupis adirect sum of abelian groups $R_{i}$ such that⁠ $R_{i}R_{j}\subseteq R_{i+j}$ ⁠.The index set is usually the set of nonnegativeintegersor the set of integers, but can be anymonoid.The direct sum decomposition is usually referred to asgradationorgrading.

Agraded moduleis defined similarly (see below for the precise definition). It generalizesgraded vector spaces.A graded module that is also a graded ring is called agraded algebra.A graded ring could also be viewed as a graded⁠ $\mathbb {Z}$ ⁠-algebra.

Theassociativityis not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies tonon-associative algebrasas well; e.g., one can consider agraded Lie algebra.

First properties

Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article.

A graded ring is aringthat is decomposed into adirect sum

R=\bigoplus _{n=0}^{\infty }R_{n}=R_{0}\oplus R_{1}\oplus R_{2}\oplus \cdots

of additive groups,such that

R_{m}R_{n}\subseteq R_{m+n}

for all nonnegative integers $m$ and⁠ $n$ ⁠.

A nonzero element of $R_{n}$ is said to behomogeneousofdegree⁠ $n$ ⁠.By definition of a direct sum, every nonzero element $a$ of $R$ can be uniquely written as a sum $a=a_{0}+a_{1}+\cdots +a_{n}$ where each $a_{i}$ is either 0 or homogeneous of degree⁠ $i$ ⁠.The nonzero $a_{i}$ are thehomogeneous componentsof⁠ $a$ ⁠.

Some basic properties are:

$R_{0}$ is asubringof⁠ $R$ ⁠;in particular, the multiplicative identity $1$ is a homogeneous element of degree zero.
For any $n$ , $R_{n}$ is a two-sided⁠ $R_{0}$ ⁠-module,and the direct sum decomposition is a direct sum of⁠ $R_{0}$ ⁠-modules.
$R$ is anassociative⁠ $R_{0}$ ⁠-algebra.

Anideal $I\subseteq R$ ishomogeneous,if for every⁠ $a\in I$ ⁠,the homogeneous components of $a$ also belong to⁠ $I$ ⁠.(Equivalently, if it is a graded submodule of⁠ $R$ ⁠;see§ Graded module.) Theintersectionof a homogeneous ideal $I$ with $R_{n}$ is an⁠ $R_{0}$ ⁠-submoduleof $R_{n}$ called thehomogeneous partof degree $n$ of⁠ $I$ ⁠.A homogeneous ideal is the direct sum of its homogeneous parts.

If $I$ is a two-sided homogeneous ideal in⁠ $R$ ⁠,then $R/I$ is also a graded ring, decomposed as

R/I=\bigoplus _{n=0}^{\infty }R_{n}/I_{n},

where $I_{n}$ is the homogeneous part of degree $n$ of⁠ $I$ ⁠.

Basic examples

Any (non-graded) ringRcan be given a gradation by letting⁠ $R_{0}=R$ ⁠,and $R_{i}=0$ fori≠ 0. This is called thetrivial gradationonR.
Thepolynomial ring $R=k[t_{1},\ldots,t_{n}]$ is graded bydegree:it is a direct sum of $R_{i}$ consisting ofhomogeneous polynomialsof degreei.
LetSbe the set of all nonzero homogeneous elements in a gradedintegral domainR.Then thelocalizationofRwith respect toSis a $\mathbb {Z}$ -graded ring.
IfIis an ideal in acommutative ringR,then ${\textstyle \bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}}$ is a graded ring called theassociated graded ringofRalongI;geometrically, it is thecoordinate ringof thenormal conealong thesubvarietydefined byI.
LetXbe atopological space,Hⁱ(X;R) theithcohomology groupwith coefficients in a ringR.ThenH^*(X;R), thecohomology ringofXwith coefficients inR,is a graded ring whose underlyinggroupis ${\textstyle \bigoplus _{i=0}^{\infty }H^{i}(X;R)}$ with the multiplicative structure given by thecup product.

Graded module

The corresponding idea inmodule theoryis that of agraded module,namely a leftmoduleMover a graded ringRsuch that

M=\bigoplus _{i\in \mathbb {N} }M_{i},

and

R_{i}M_{j}\subseteq M_{i+j}

for every $i$ and $j$ .

Example:agraded vector spaceis an example of a graded module over afield(with the field having trivial grading).

Example:a graded ring is a graded module over itself. An ideal in a graded ring is homogeneousif and only ifit is a graded submodule. Theannihilatorof a graded module is a homogeneous ideal.

Example:Given an idealIin a commutative ringRand anR-moduleM,the direct sum $\bigoplus _{n=0}^{\infty }I^{n}M/I^{n+1}M$ is a graded module over the associated graded ring ${\textstyle \bigoplus _{0}^{\infty }I^{n}/I^{n+1}}$ .

Amorphism $f:N\to M$ of graded modules, called agraded morphismorgraded homomorphism,is ahomomorphismof the underlying modules that respects grading; i.e.,⁠ $f(N_{i})\subseteq M_{i}$ ⁠.Agraded submoduleis a submodule that is a graded module in own right and such that the set-theoreticinclusionis a morphism of graded modules. Explicitly, a graded moduleNis a graded submodule ofMif and only if it is a submodule ofMand satisfies⁠ $N_{i}=N\cap M_{i}$ ⁠.Thekerneland theimageof a morphism of graded modules are graded submodules.

Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in thecenteris the same as to give the structure of a graded algebra to the latter ring.

Given a graded module $M$ ,the $\ell$ -twist of $M$ is a graded module defined by $M(\ell )_{n}=M_{n+\ell }$ (cf.Serre's twisting sheafinalgebraic geometry).

LetMandNbe graded modules. If $f\colon M\to N$ is a morphism of modules, thenfis said to have degreedif $f(M_{n})\subseteq N_{n+d}$ .Anexterior derivativeofdifferential formsindifferential geometryis an example of such a morphism having degree 1.

Invariants of graded modules

Given a graded moduleMover a commutative graded ringR,one can associate theformal power series⁠ $P(M,t)\in \mathbb {Z} [\![t]\!]$ ⁠:

P(M,t)=\sum \ell (M_{n})t^{n}

(assuming $\ell (M_{n})$ are finite.) It is called theHilbert–Poincaré seriesofM.

A graded module is said to be finitely generated if the underlying module isfinitely generated.The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)

SupposeRis apolynomial ring⁠ $k[x_{0},\dots,x_{n}]$ ⁠,ka field, andMa finitely generated graded module over it. Then the function $n\mapsto \dim _{k}M_{n}$ is called the Hilbert function ofM.The function coincides with theinteger-valued polynomialfor largencalled theHilbert polynomialofM.

Graded algebra

Anassociative algebraAover a ringRis agraded algebraif it is graded as a ring.

In the usual case where the ringRis not graded (in particular ifRis a field), it is given the trivial grading (every element ofRis of degree 0). Thus, $R\subseteq A_{0}$ and the graded pieces $A_{i}$ areR-modules.

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

R_{i}A_{j}\subseteq A_{i+j}

In other words, we requireAto be a graded left module overR.

Examples of graded algebras are common in mathematics:

Polynomial rings.The homogeneous elements of degreenare exactly the homogeneous polynomials of degreen.
Thetensor algebra $T^{\bullet }V$ of avector spaceV.The homogeneous elements of degreenare thetensorsof ordern,⁠ $T^{n}V$ ⁠.
Theexterior algebra $\textstyle \bigwedge \nolimits ^{\bullet }V$ and thesymmetric algebra $S^{\bullet }V$ are also graded algebras.
Thecohomology ring $H^{\bullet }$ in anycohomology theoryis also graded, being the direct sum of the cohomology groups⁠ $H^{n}$ ⁠.

Graded algebras are much used incommutative algebraandalgebraic geometry,homological algebra,andalgebraic topology.One example is the close relationship betweenhomogeneous polynomialsandprojective varieties(cf.Homogeneous coordinate ring.)

G-graded rings and algebras

The above definitions have been generalized to rings graded using anymonoidGas an index set. AG-graded ringRis a ring with a direct sum decomposition

R=\bigoplus _{i\in G}R_{i}

such that

R_{i}R_{j}\subseteq R_{i\cdot j}.

Elements ofRthat lie inside $R_{i}$ for some $i\in G$ are said to behomogeneousofgradei.

The previously defined notion of "graded ring" now becomes the same thing as an $\mathbb {N}$ -graded ring, where $\mathbb {N}$ is the monoid ofnatural numbersunder addition. The definitions for graded modules and algebras can also be extended this way replacing the inde xing set $\mathbb {N}$ with any monoidG.

Remarks:

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

Examples:

Agroupnaturally grades the correspondinggroup ring;similarly,monoid ringsare graded by the corresponding monoid.
An (associative)superalgebrais another term for a $\mathbb {Z} _{2}$ -graded algebra. Examples includeClifford algebras.Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

Anticommutativity

Some graded rings (or algebras) are endowed with ananticommutativestructure. This notion requires ahomomorphismof the monoid of the gradation into the additive monoid of $\mathbb {Z} /2\mathbb {Z}$ ,the field with two elements. Specifically, asigned monoidconsists of a pair $(\Gamma,\varepsilon )$ where $\Gamma$ is a monoid and $\varepsilon \colon \Gamma \to \mathbb {Z} /2\mathbb {Z}$ is a homomorphism of additive monoids. Ananticommutative $\Gamma$ -graded ringis a ringAgraded with respect to $\Gamma$ such that:

xy=(-1)^{\varepsilon (\deg x)\varepsilon (\deg y)}yx,

for all homogeneous elementsxandy.

Examples

Anexterior algebrais an example of an anticommutative algebra, graded with respect to the structure $(\mathbb {Z},\varepsilon )$ where $\varepsilon \colon \mathbb {Z} \to \mathbb {Z} /2\mathbb {Z}$ is the quotient map.
Asupercommutative algebra(sometimes called askew-commutative associative ring) is the same thing as an anticommutative $(\mathbb {Z},\varepsilon )$ -graded algebra, where $\varepsilon$ is theidentity mapof the additive structure of⁠ $\mathbb {Z} /2\mathbb {Z}$ ⁠.

Graded monoid

Intuitively, a gradedmonoidis the subset of a graded ring, ${\textstyle \bigoplus _{n\in \mathbb {N} _{0}}R_{n}}$ ,generated by the $R_{n}$ 's, without using the additive part. That is, the set of elements of the graded monoid is $\bigcup _{n\in \mathbb {N} _{0}}R_{n}$ .

Formally, a graded monoid^[1]is a monoid $(M,\cdot )$ ,with a gradation function $\phi:M\to \mathbb {N} _{0}$ such that $\phi (m\cdot m')=\phi (m)+\phi (m')$ .Note that the gradation of $1_{M}$ is necessarily 0. Some authors request furthermore that $\phi (m)\neq 0$ whenmis not the identity.

Assuming the gradations of non-identity elements are non-zero, the number of elements of gradationnis at most $g^{n}$ wheregis the cardinality of agenerating setGof the monoid. Therefore the number of elements of gradationnor less is at most $n+1$ (for $g=1$ ) or ${\textstyle {\frac {g^{n+1}-1}{g-1}}}$ else. Indeed, each such element is the product of at mostnelements ofG,and only ${\textstyle {\frac {g^{n+1}-1}{g-1}}}$ such products exist. Similarly, the identity element can not be written as the product of two non-identity elements. That is, there is no unitdivisorin such a graded monoid.

Power series indexed by a graded monoid

These notions allow us to extend the notion ofpower series ring.Instead of the inde xing family being $\mathbb {N}$ ,the inde xing family could be any graded monoid, assuming that the number of elements of degreenis finite, for each integern.

More formally, let $(K,+_{K},\times _{K})$ be an arbitrarysemiringand $(R,\cdot,\phi )$ a graded monoid. Then $K\langle \langle R\rangle \rangle$ denotes the semiring of power series with coefficients inKindexed byR.Its elements are functions fromRtoK.The sum of two elements $s,s'\in K\langle \langle R\rangle \rangle$ is defined pointwise, it is the function sending $m\in R$ to $s(m)+_{K}s'(m)$ ,and the product is the function sending $m\in R$ to the infinite sum $\sum _{p,q\in R \atop p\cdot q=m}s(p)\times _{K}s'(q)$ .This sum is correctly defined (i.e., finite) because, for eachm,there are only a finite number of pairs(p,q)such thatpq=m.

Example

Informal language theory,given an Alpha betA,thefree monoidof words overAcan be considered as a graded monoid, where the gradation of a word is its length.

Notes

Citations

^Sakarovitch, Jacques (2009). "Part II: The power of algebra".Elements of automata theory.Translated by Thomas, Reuben. Cambridge University Press. p. 384.ISBN 978-0-521-84425-3.Zbl 1188.68177.

References

Lang, Serge(2002),Algebra,Graduate Texts in Mathematics,vol. 211 (Revised third ed.), New York: Springer-Verlag,ISBN 978-0-387-95385-4,MR 1878556.
Bourbaki, N.(1974). "Ch. 1–3, 3 §3".Algebra I.ISBN 978-3-540-64243-5.
Steenbrink, J. (1977)."Intersection form for quasi-homogeneous singularities"(PDF).Compositio Mathematica.34(2): 211–223 See p. 211.ISSN 0010-437X.
Matsumura, H. (1989). "5 Dimension theory §S3 Graded rings, the Hilbert function and the Samuel function".Commutative Ring Theory.Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press.ISBN 978-1-107-71712-1.

[1] Sakarovitch, Jacques (2009). "Part II: The power of algebra".Elements of automata theory.Translated by Thomas, Reuben. Cambridge University Press. p. 384.ISBN 978-0-521-84425-3.Zbl 1188.68177.

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