Associated graded ring

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 + 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^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 \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}_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) = \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_jI_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.

References

References

1. {{harvnb. Eisenbud. 1995
2. {{harvnb. Zariski. Samuel. 1975