“Associated graded ring”的意思、由来-开放百科全书

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Associated graded ring

释义

Basic definitions and properties
gr of a quotient module
Examples
Generalization to multiplicative filtrations
See also
References

In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:

.

Similarly, if M is a left R-module, then the associated graded module is the graded module over :

.

Basic definitions and properties

For a ring R and ideal I, multiplication in is defined as follows: First, consider homogeneous elements and and suppose is a representative of a and is a representative of b. Then define to be the equivalence class of in . Note that this is well-defined modulo . Multiplication of inhomogeneous elements is defined by using the distributive property.

A ring or module may be related to its associated graded ring or module through the initial form map. Let M be an R-module and I an ideal of R. Given , the initial form of f in , written , is the equivalence class of f in where m is the maximum integer such that . If for every m, then set . The initial form map is only a map of sets and generally not a homomorphism. For a submodule , is defined to be the submodule of generated by . This may not be the same as the submodule of generated by the only initial forms of the generators of N.

A ring inherits some "good" properties from its associated graded ring. For example, if R is a noetherian local ring, and is an integral domain, then R is itself an integral domain.^[1]

gr of a quotient module

Let be left modules over a ring R and I an ideal of R. Since

(the last equality is by modular law), there is a canonical identification:^[2]

where

called the submodule generated by the initial forms of the elements of .

Examples

Let U be the enveloping algebra of a Lie algebra over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that is a polynomial ring; in fact, it is the coordinate ring .

The associated graded algebra of a Clifford algebra is an exterior algebra; i.e., a Clifford algebra degenerates to an exterior algebra.

Generalization to multiplicative filtrations

The associated graded can also be defined more generally for multiplicative descending filtrations of R (see also filtered ring.) Let F be a descending chain of ideals of the form

such that . The graded ring associated with this filtration is . Multiplication and the initial form map are defined as above.

References

1. ^{{harvnb|Eisenbud|loc=Corollary 5.5}}
2. ^{{harvnb|Zariski–Samuel|loc=Ch. VIII, a paragraph after Theorem 1.}}

{{cite book|last=Eisenbud|first=David|authorlink=David Eisenbud|title=Commutative Algebra|series=Graduate Texts in Mathematics|volume=150|publisher=Springer-Verlag|year=1995|isbn=0-387-94268-8|doi=10.1007/978-1-4612-5350-1|mr=1322960|location=New York}}
{{cite book|last=Matsumura|first=Hideyuki|title=Commutative ring theory|others=Translated from the Japanese by M. Reid|edition=Second|series=Cambridge Studies in Advanced Mathematics|volume=8|publisher=Cambridge University Press|location=Cambridge|year=1989|isbn=0-521-36764-6|mr=1011461}}
{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90171-8 |mr=0389876 | year=1975}}

1 : Ring theory

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Basic definitions and properties

gr of a quotient module

Examples

Generalization to multiplicative filtrations

See also

References