Associated graded ring
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.
gr of a quotient module
Let be left modules over a ring R and I an ideal of R. Since, there is a canonical identification:
where
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 Let F be a descending chain of ideals of the formsuch that. The graded ring associated with this filtration is. Multiplication and the initial form map are defined as above.