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
![{\displaystyle {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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/718859a3ae92c08f2d4e34a65273d0dcc1cb41c4)
(the last equality is by modular law), there is a canonical identification:[2]
![{\displaystyle \operatorname {gr} _{I}(M/N)=\operatorname {gr} _{I}M/\operatorname {in} (N)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1d76d5fbe066228669ff2873bb3a26f7b4a01b)
where
![{\displaystyle \operatorname {in} (N)=\bigoplus _{n=0}^{\infty }{I^{n}M\cap N+I^{n+1}M \over I^{n+1}M},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead070713da21c178f1e8e728e41b0cf59ff08de)
called the submodule generated by the initial forms of the elements of
.
Examples
Let U be the universal 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
![{\displaystyle R=I_{0}\supset I_{1}\supset I_{2}\supset \dotsb }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4711a2db9747bc68978af91f54e268f9c4dac9)
such that
. The graded ring associated with this filtration is
. Multiplication and the initial form map are defined as above.