Graded algebra

In, in particular , a graded ring is a that is a  $$R_i$$ such that $$R_i R_j \subseteq R_{i+j}$$. The index set is usually the set of nonnegative integers or the set of integers, but can be any. The direct sum decomposition is usually referred to as gradation or grading.

A graded module is defined similarly (see below for the precise definition). It generalizes s. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded Z-algebra.

The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to s as well; e.g., one can consider a.

First properties
Let
 * $$R = \bigoplus_{n\in \mathbb N_0}R_n = R_0 \oplus R_1 \oplus R_2 \oplus \cdots$$

be a graded ring. Elements of any factor $$R_n$$ of the decomposition are called homogeneous elements of degree n. Every nonzero element a of R may be uniquely written as a sum a = a1 + a2 + ... + an with all ai homogeneous elements of distinct Ri. These ai are called the homogeneous components of a.

Some basic properties are:
 * $$R_0$$ is a subring of R; in particular, the additive identity 0 and the multiplicative identity 1 are homogeneous elements of degree zero.
 * A commutative $$\mathbb{N}_0$$-graded ring $$R=\bigoplus_{i=0}^\infty R_i$$ is a if and only if $$R_0$$ is Noetherian and R is finitely generated as an algebra over $$R_0$$.

An $$I\subseteq R$$ is homogeneous if, for every element $$a \in I$$, its homogeneous components belong also to $$I.$$ (Equivalently, they are graded submodules of R; see .) The intersections of a homogeneous ideal $$I$$ with the $$R_i$$ are called the homogeneous parts of $$I$$. A homogeneous ideal is the direct sum of its homogeneous parts.

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


 * $$R/I = \bigoplus_{n\in \mathbb N}(R_n + I)/I.$$

Basic examples

 * Any (non-graded) ring R can be given a gradation by letting $$R_0=R$$, and $$R_i=0$$ for i ≠ 0. This is called the trivial gradation on R.
 * The $$R = k[t_1, \ldots, t_n]$$ is graded by : it is a direct sum of $$R_i$$ consisting of s of degree i.
 * Let S be the set of all nonzero homogeneous elements in a graded R. Then the  of R with respect to S is a Z-graded ring.
 * If I is an ideal in a commutative ring R, then $$\oplus_0^{\infty} I^n/I^{n+1}$$ is a graded ring called the of R along I; geometrically, it is the coordinate ring of the  along the subvariety defined by I.

Graded module
The corresponding idea in is that of a graded module, namely a left  M over a graded ring R such that also
 * $$M = \bigoplus_{i\in \mathbb{N}_0}M_i ,$$

and
 * $$R_iM_j \subseteq M_{i+j}.$$

Example: a is an example of a graded module over a field (with the field having trivial grading).

Example: a graded ring is a graded module over itself. An ideal in a graded ring is homogeneous if and only if it is a graded submodule. The of a graded module is a homogeneous ideal.

Example: Given an ideal I in a commutative ring R and an R-module M, $$\bigoplus_{n=0}^{\infty} I^n M/I^{n+1} M$$ is a graded module over the associated graded ring $$\oplus_0^{\infty} I^n/I^{n+1}$$.

A morphism $$f: N \to M$$ between graded modules, called a graded morphism, is a morphism of underlying modules that respects grading; i.e., $$f(N_i) \subseteq M_i$$. A graded submodule is a submodule that is a graded module in own right and such that the set-theoretic inclusion is a morphism of graded modules. Explicitly, a graded module N is a graded submodule of M if and only if it is a submodule of M and satisfies $$N_i = N \cap M_i$$. The kernel and the image of a morphism of graded modules are graded submodules.

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

Given a graded module M, the ℓ-twist of $$M(\ell)$$ is a graded module defined by $$M(\ell)_n = M_{n+\ell}$$. (cf. in algebraic geometry.)

Let M and N be graded modules. If $$f: M \to N$$ is a morphism of modules, then f is said to have degree d if $$f(M_n) \subseteq N_{n+d}$$. An of differential forms in differential geometry is an example of such a morphism having degree 1.

Invariants of graded modules
Given a graded module M over a commutative graded ring R, one can associate the formal 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 the of M.

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

Suppose R is a polynomial ring $$k[x_0, \dots, x_n]$$, k a field, and M a finitely generated graded module over it. Then the function $$n \mapsto \dim_k M_n$$ is called the Hilbert function of M. The function coincides with the for large n called the  of M.

Graded algebra
An 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 degree 0). Thus, R ⊆ R0 and the Ai are R-modules.

In the case where the ring R is also a graded ring, then one requires that
 * $$R_iA_j \subseteq A_{i+j}$$

In other words, we require A to be a graded left module over R.

Examples of graded algebras are common in mathematics:


 * s. The homogeneous elements of degree n are exactly the homogeneous s of degree n.
 * The T•V of a  V. The homogeneous elements of degree n are the s of order n, TnV.
 * The Λ•V and  S•V are also graded algebras.
 * The H• in any  is also graded, being the direct sum of the Hn.

Graded algebras are much used in and,  and. One example is the close relationship between homogeneous s and. (cf. .)

G-graded rings and algebras
The above definitions have been generalized to rings graded using any G as an index set. A G-graded ring R is 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 of R that lie inside $$R_i$$ for some $$i \in G$$ are said to be homogeneous of grade i.

The previously defined notion of "graded ring" now becomes the same thing as an N-graded ring, where N is the monoid of 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, s may replace s.

Examples:
 * A group naturally grades the corresponding ; similarly, s are graded by the corresponding monoid.
 * An (associative) is another term for a -graded algebra. Examples include s. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

Anticommutativity
Some graded rings (or algebras) are endowed with an structure. This notion requires a of the monoid of the gradation into the additive monoid of Z/2Z, the field with two elements. Specifically, a signed monoid consists of a pair (Γ, ε) where Γ is a monoid and ε : Γ → Z/2Z is a homomorphism of additive monoids. An anticommutative Γ-graded ring is a ring A graded with respect to Γ such that:
 * $$xy=(-1)^{\varepsilon (\deg x) \varepsilon (\deg y)}yx ,$$

for all homogeneous elements x and y.

Examples

 * An is an example of an anticommutative algebra, graded with respect to the structure (Z, ε) where ε : Z → Z/2Z is the quotient map.
 * A (sometimes called a skew-commutative associative ring) is the same thing as an anticommutative (Z/2Z, ε)-graded algebra, where ε is the identity  of the additive structure of Z/2Z.

Graded monoid
Intuitively, a graded is the subset of a graded ring, $$\bigoplus_{n\in \mathbb N_0}R_n$$, 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 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 $$\phi(m)\ne 0$$ when m is not the identity.

Assuming the gradations of non-identity elements are non zero, the number of elements of gradation n is at most $$\frac{g^{n+1}-1}{g-1}$$ where g is the cardinality of a G of the monoid. Indeed, each such element is the product of at most n elements of G, and only $$\frac{g^{n+1}-1}{g-1}$$ such products exist. Similarly, the identity element can not be written as the product of two non-identity elements. That is, there is no unit divisor in such a graded monoid.

Power series indexed by a graded monoid
This notions allows to extends the notion of. Instead of having the indexing family being $$\mathbb N$$, the indexing family could be any graded monoid, assuming that the number of elements of degree n is finite, for each integer n.

More formally, let $$(K,+_K,\times_K)$$ be an arbitrary and $$(R,\cdot,\phi)$$ a graded monoid. Then $$K\langle\langle R\rangle\rangle$$ denotes the semiring of power series with coefficients in K indexed by R. Its elements are functions from R to K. The sum of two elements $$s,s'\in K\langle\langle R\rangle\rangle$$ is defined point-wise, it is the function sending $$m\in R$$ to $$s(m)+_Ks'(m)$$. And the product is the function sending $$m\in R$$ to the infinite sum $$\sum_{p,q\in R,p\cdot q=m}s(p)\times_K s'(q)$$. This sum is correctly defined (i.e., finite) because, for each m, only a finite number of pairs (p, q) such that pq = m exist.

Example
In, given an alphabet A, the of words over A can be considered as a graded monoid, where the gradation of a word is its length.