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