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 $$\operatorname{gr}_IR$$ 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.