Associated Graded Ring

In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring:
:$\operatorname_I R = \oplus_^\infty I^n/I^$.
Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the graded module over $\operatorname_I R$:
:$\operatorname_I M = \oplus_^\infty I^n M/ I^ M$.

Basic definitions and properties

For a ring ''R'' and ideal ''I'', multiplication in $\operatorname_IR$ is defined as follows: First, consider homogeneous elements $a \in I^i/I^$ and $b \in I^j/I^$ 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 $I^/I^$. Note that this is well-defined modulo $I^$. 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 $\operatorname_I M$, written $\mathrm(f)$, is the equivalence class of ''f'' in $I^mM/I^M$ where ''m'' is the maximum integer such that $f\in I^mM$. If $f \in I^mM$ for every ''m'', then set $\mathrm(f) = 0$. The initial form map is only a map of sets and generally not a homomorphism. For a submodule $N \subset M$, $\mathrm(N)$ is defined to be the submodule of $\operatorname_I M$ generated by $\$. This may not be the same as the submodule of $\operatorname_IM$ 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 $\operatorname_I R$ 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
:$\simeq \simeq =$
(the last equality is by modular law), there is a canonical identification:
:$\operatorname_I (M/N) = \operatorname_I M / \operatorname(N)$
where
:$\operatorname(N) = \bigoplus_^ ,$
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 $\mathfrak$ over a field ''k''; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that $\operatorname U$ is a polynomial ring; in fact, it is the coordinate ring k mathfrak^*/math>.

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 $I_jI_k \subset I_$. The graded ring associated with this filtration is $\operatorname_F R = \bigoplus_^\infty I_n/ I_$. Multiplication and the initial form map are defined as above.

See also

* Graded (mathematics)
* Rees algebra

References

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* {{Citation , last1=Zariski , first1=Oscar , author1-link=Oscar Zariski , last2=Samuel , first2=Pierre , author2-link=Pierre Samuel , title=Commutative algebra. Vol. II , publisher=Springer-Verlag , location=Berlin, New York , isbn=978-0-387-90171-8 , mr=0389876 , year=1975

Ring theory