Associated graded ring

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

$\operatorname {gr} _{I}R=\bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}$.

Similarly, if M is a left R-module, then the associated graded module is the graded module over $\operatorname {gr} _{I}R$:

$\operatorname {gr} _{I}M=\bigoplus _{n=0}^{\infty }I^{n}M/I^{n+1}M$.

Basic definitions and properties

For a ring R and ideal I, multiplication in $\operatorname {gr} _{I}R$ is defined as follows: First, consider homogeneous elements $a\in I^{i}/I^{i+1}$ and $b\in I^{j}/I^{j+1}$ 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 $I^{i+j}/I^{i+j+1}$. Note that this is well-defined modulo $I^{i+j+1}$. 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 $\mathrm {in} (f)$, is the equivalence class of f in $I^{m}M/I^{m+1}M$ where m is the maximum integer such that $f\in I^{m}M$. If $f\in I^{m}M$ for every m, then set $\mathrm {in} (f)=0$. The initial form map is only a map of sets and generally not a homomorphism. For a submodule $N\subset M$, $\mathrm {in} (N)$ is defined to be the submodule of $\operatorname {gr} _{I}M$ generated by $\{\mathrm {in} (f)|f\in N\}$. This may not be the same as the submodule of $\operatorname {gr} _{I}M$ 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

$\operatorname {in} (N)=\bigoplus _{n=0}^{\infty }{I^{n}M\cap N+I^{n+1}M \over I^{n+1}M},$

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 ${\mathfrak {g}}$ 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 $k[{\mathfrak {g}}^{*}]$.

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

$R=I_{0}\supset I_{1}\supset I_{2}\supset \dotsb$

such that $I_{j}I_{k}\subset I_{j+k}$. The graded ring associated with this filtration is $\operatorname {gr} _{F}R=\bigoplus _{n=0}^{\infty }I_{n}/I_{n+1}$. Multiplication and the initial form map are defined as above.

See also

• Graded (mathematics)
• Rees algebra

References

• Eisenbud, David (1995). Commutative Algebra. Graduate Texts in Mathematics. Vol. 150. New York: Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8. MR 1322960.
• Matsumura, Hideyuki (1989). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated from the Japanese by M. Reid (Second ed.). Cambridge: Cambridge University Press. ISBN 0-521-36764-6. MR 1011461.
• Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876