Graded ring

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In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. 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 graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded -algebra.

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

First propertiesEdit

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

A graded ring is a ring that is decomposed into a direct sum

 

of additive groups, such that

 

for every nonnegative integers m and n.

A nonzero element of   is said to be homogeneous of degree n. By definition of a direct sum, every nonzero element a of R can be uniquely written as a sum   where each   is either 0 or homogeneous of degree i. The nonzero   are the homogeneous components of a.

Some basic properties are:

  •   is a subring of R; in particular, the multiplicative identity 1 is an homogeneous element of degree zero.
  • For any n,   is a two-sided  -module, and the direct sum decomposition is a direct sum of  -modules.
  • R is an associative  -algebra.

An ideal   is homogeneous if, for every  , its homogeneous components belong also to   (Equivalently, the ideal is a graded submodules of R; see § Graded module.) The intersections of a homogeneous ideal   with   is a  -submodule called the homogeneous part of degree n of  . A homogeneous ideal is the direct sum of its homogeneous parts.

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

 

where   is the homogeneous part of degree n of I.

Basic examplesEdit

  • Any (non-graded) ring R can be given a gradation by letting  , and   for i ≠ 0. This is called the trivial gradation on R.
  • The polynomial ring   is graded by degree: it is a direct sum of   consisting of homogeneous polynomials of degree i.
  • Let S be the set of all nonzero homogeneous elements in a graded integral domain R. Then the localization of R with respect to S is a  -graded ring.
  • If I is an ideal in a commutative ring R, then   is a graded ring called the associated graded ring of R along I; geometrically, it is the coordinate ring of the normal cone along the subvariety defined by I.

Graded moduleEdit

The corresponding idea in module theory is that of a graded module, namely a left module M over a graded ring R such that also

 

and

 

Example: a graded vector space 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 annihilator of a graded module is a homogeneous ideal.

Example: Given an ideal I in a commutative ring R and an R-module M, the direct sum   is a graded module over the associated graded ring  .

A morphism   between graded modules, called a graded morphism, is a morphism of underlying modules that respects grading; i.e.,  . 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  . 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 another 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  , the  -twist of   is a graded module defined by  . (cf. Serre's twisting sheaf in algebraic geometry.)

Let M and N be graded modules. If   is a morphism of modules, then f is said to have degree d if  . An exterior derivative of differential forms in differential geometry is an example of such a morphism having degree 1.

Invariants of graded modulesEdit

Given a graded module M over a commutative graded ring R, one can associate the formal power series  :

 

(assuming   are finite.) It is called the Hilbert–Poincaré series 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 a field, and M a finitely generated graded module over it. Then the function   is called the Hilbert function of M. The function coincides with the integer-valued polynomial for large n called the Hilbert polynomial of M.

Graded algebraEdit

An algebra 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,   and the graded pieces   are R-modules.

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

 

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

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   of a vector space V. The homogeneous elements of degree n are the tensors of order n,  .
  • The exterior algebra   and the symmetric algebra   are also graded algebras.
  • The cohomology ring   in any cohomology theory is also graded, being the direct sum of the cohomology groups  .

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 (cf. homogeneous coordinate ring.)

G-graded rings and algebrasEdit

The above definitions have been generalized to rings graded using any monoid G as an index set. A G-graded ring R is a ring with a direct sum decomposition

 

such that

 

Elements of R that lie inside   for some   are said to be homogeneous of grade i.

The previously defined notion of "graded ring" now becomes the same thing as an  -graded ring, where   is the monoid of non-negative integers under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set   with any monoid G.

Remarks:

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

Examples:

AnticommutativityEdit

Some graded rings (or algebras) are endowed with an anticommutative structure. This notion requires a homomorphism of the monoid of the gradation into the additive monoid of  , the field with two elements. Specifically, a signed monoid consists of a pair   where   is a monoid and   is a homomorphism of additive monoids. An anticommutative  -graded ring is a ring A graded with respect to Γ such that:

 

for all homogeneous elements x and y.

ExamplesEdit

  • An exterior algebra is an example of an anticommutative algebra, graded with respect to the structure   where   is the quotient map.
  • A supercommutative algebra (sometimes called a skew-commutative associative ring) is the same thing as an anticommutative  -graded algebra, where   is the identity endomorphism of the additive structure of  .

Graded monoidEdit

Intuitively, a graded monoid is the subset of a graded ring,  , generated by the  's, without using the additive part. That is, the set of elements of the graded monoid is  .

Formally, a graded monoid[1] is a monoid  , with a gradation function   such that  . Note that the gradation of   is necessarily 0. Some authors request furthermore that   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   where g is the cardinality of a generating set G of the monoid. Therefore the number of elements of gradation n or less is at most   (for  ) or   else. Indeed, each such element is the product of at most n elements of G, and only   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 monoidEdit

This notions allows to extends the notion of power series ring. Instead of having the indexing family being  , 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   be an arbitrary semiring and   a graded monoid. Then   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   is defined point-wise, it is the function sending   to  . And the product is the function sending   to the infinite sum  . 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.

ExampleEdit

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

See alsoEdit

ReferencesEdit

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