Associated graded ring

In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:

${\mathrm{gr}}_{I}R=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{I}^n/I^{n+1}$.

Similarly, if M is a left R-module, then the associated graded module is the graded module over ${\mathrm{gr}}_{I}R$:

${\mathrm{gr}}_{I}M=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{I}^{n}M/I^{n+1}M$.

Basic definitions and properties

For a ring R and ideal I, multiplication in ${\mathrm{gr}}_{I}R$ 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 $a^{\prime }\in I^i$ is a representative of a and $b^{\prime }\in I^j$ is a representative of b. Then define $ab$ to be the equivalence class of $a^{\prime }{b}^{\prime }$ 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 ${\mathrm{gr}}_{I}M$, written $\mathrm{i}\mathrm{n}(f)$, is the equivalence class of f in $I^{m}M/I^{m+1}M$ where m is the maximum integer such that $f\in I^{m}M$. If $f\in I^{m}M$ for every m, then set $\mathrm{i}\mathrm{n}(f)=0$. The initial form map is only a map of sets and generally not a homomorphism. For a submodule $N\subset M$, $\mathrm{i}\mathrm{n}(N)$ is defined to be the submodule of ${\mathrm{gr}}_{I}M$ generated by $\{\mathrm{i}\mathrm{n}(f)|f\in N\}$. This may not be the same as the submodule of ${\mathrm{gr}}_{I}M$ 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 ${\mathrm{gr}}_{I}R$ 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

$\frac{I^n(M/N)}{I^{n+1}(M/N)}\simeq \frac{I^{n}M+N}{I^{n+1}M+N}\simeq \frac{I^{n}M}{I^{n}M\cap (I^{n+1}M+N)}=\frac{I^{n}M}{I^{n}M\cap N+I^{n+1}M}$

(the last equality is by modular law), there is a canonical identification:

${\mathrm{gr}}_I(M/N)={\mathrm{gr}}_{I}M/\mathrm{in}(N)$

where

$\mathrm{in}(N)=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}\frac{I^{n}M\cap N+I^{n+1}M}{I^{n+1}M},$

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 $𝔤$ over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that $\mathrm{gr}U$ is a polynomial ring; in fact, it is the coordinate ring $k[{𝔤}^{\ast }]$.

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

$R=I_0\supset I_1\supset I_2\supset \cdots$

such that $I_j{I}_k\subset I_{j+k}$. The graded ring associated with this filtration is ${\mathrm{gr}}_{F}R=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{I}_n/I_{n+1}$. Multiplication and the initial form map are defined as above.

See also

• Graded (mathematics)
• Rees algebra

References

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