Associated graded ring explained

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

\operatorname{gr} _{I} R = o plus ^{inf ty} _{n =0} I ^{n/I} ^{n+1} .Similarly, if M is a left R-module, then the associated graded module is the graded module over \operatorname{gr} _{I} R :\operatorname{gr} _{I} M = o plus ^{inf ty} _{n =0} I ^{n} M/ I ^{n+1} M .

Basic definitions and properties

For a ring R and ideal I, multiplication in

\operatorname{gr} _{IR} is defined as follows: First, consider homogeneous elements a \in I ^{i/I} ^{i} and b \in I ^{j/I} ^{j} and suppose a' \in I ^{i} is a representative of a and b' \in I ^{j} is a representative of b. Then define ab to be the equivalence class of a'b' in I ^{i} /I ^{i} . Note that this is well-defined modulo I ^{i} . 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

f \in M , the initial form of f in \operatorname{gr} _{I} M , written in (f) , is the equivalence class of f in I ^{mM/I} ^{m+1} M where m is the maximum integer such that f\in I ^{mM} . If f \in I ^{mM} for every m, then set in (f) = 0 . The initial form map is only a map of sets and generally not a homomorphism. For a submodule N \subset M , in (N) is defined to be the submodule of \operatorname{gr} _{I} M generated by \{ in (f) | f \in N\} . This may not be the same as the submodule of \operatorname{gr} _{IM} 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

\operatorname{gr} _{I} R is an integral domain, then R is itself an integral domain.

gr of a quotient module

Let

N \subset M be left modules over a ring R and I an ideal of R. Since{I ^{n(M/N)} \over I ^{n+1} (M/N)} \simeq {I ^{n} M + N \over I ^{n+1} M + N} \simeq {I ^{n} M \over I ^{n} M \cap (I ^{n+1} M + N)} = {I ^{n} M \over I ^{n} M \cap N + I ^{n+1} M} (the last equality is by modular law), there is a canonical identification:\operatorname{gr} _{I} (M/N) = \operatorname{gr} _{I} M / \operatorname{in}(N) where\operatorname{in}(N) = o plus ^{inf ty} _{n =0} {I ^{n} M \cap N + I ^{n+1} M \over I ^{n+1} M}, called the submodule generated by the initial forms of the elements of N .

Examples

Let U be the universal enveloping algebra of a Lie algebra

ak{g} over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that \operatorname{gr} U is a polynomial ring; in fact, it is the coordinate ring k[ ak{g} ^{*]} .

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

R = I _{0} \supset I _{1} \supset I _{2} \supset ... b such that I _{jI} _{k} \subset I _{j} . The graded ring associated with this filtration is \operatorname{gr} _{F} R = o plus ^{inf ty} _{n =0} I _{n/} I _{n+1} . Multiplication and the initial form map are defined as above.

See also

• Graded (mathematics)
• Rees algebra

References

• Book: Eisenbud, David. David Eisenbud

. David Eisenbud. Commutative Algebra. Graduate Texts in Mathematics. 150. Springer-Verlag. 1995. 0-387-94268-8. 10.1007/978-1-4612-5350-1. 1322960. New York.

• Book: Matsumura, Hideyuki. Commutative ring theory. Translated from the Japanese by M. Reid. Second. Cambridge Studies in Advanced Mathematics. 8. Cambridge University Press. Cambridge. 1989. 0-521-36764-6. 1011461.