Associated graded ring

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

${\displaystyle \operatorname {gr} _{I}R=\oplus _{n=0}^{\infty }I^{n}/I^{n+1}}$.

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

${\displaystyle \operatorname {gr} _{I}M=\oplus _{n=0}^{\infty }I^{n}M/I^{n+1}M}$.

Basic definitions and properties

For a ring R and ideal I, multiplication in ${\displaystyle \operatorname {gr} _{I}R}$ is defined as follows: First, consider homogeneous elements ${\displaystyle a\in I^{i}/I^{i+1}}$ and ${\displaystyle b\in I^{j}/I^{j+1}}$ and suppose ${\displaystyle a'\in I^{i}}$ is a representative of a and ${\displaystyle b'\in I^{j}}$ is a representative of b. Then define ${\displaystyle ab}$ to be the equivalence class of ${\displaystyle a'b'}$ in ${\displaystyle I^{i+j}/I^{i+j+1}}$. Note that this is well-defined modulo ${\displaystyle I^{i+j+1}}$. 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 ${\displaystyle f\in M}$, the initial form of f in ${\displaystyle \operatorname {gr} _{I}M}$, written ${\displaystyle \mathrm {in} (f)}$, is the equivalence class of f in ${\displaystyle I^{m}M/I^{m+1}M}$ where m is the maximum integer such that ${\displaystyle f\in I^{m}M}$. If ${\displaystyle f\in I^{m}M}$ for every m, then set ${\displaystyle \mathrm {in} (f)=0}$. The initial form map is only a map of sets and generally not a homomorphism. For a submodule ${\displaystyle N\subset M}$, ${\displaystyle \mathrm {in} (N)}$ is defined to be the submodule of ${\displaystyle \operatorname {gr} _{I}M}$ generated by ${\displaystyle \{\mathrm {in} (f)|f\in N\}}$. This may not be the same as the submodule of ${\displaystyle \operatorname {gr} _{I}M}$ 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 ${\displaystyle \operatorname {gr} _{I}R}$ is an integral domain, then R is itself an integral domain. ^[1]

gr of a quotient module

Let ${\displaystyle N\subset M}$ be left modules over a ring R and I an ideal of R. Since

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

(the last equality is by modular law), there is a canonical identification: ^[2]

${\displaystyle \operatorname {gr} _{I}(M/N)=\operatorname {gr} _{I}M/\operatorname {in} (N)}$

where

${\displaystyle \operatorname {in} (N)=\bigoplus _{n=0}^{\infty }{I^{n}M\cap N+I^{n+1}M \over I^{n+1}M},}$

called the submodule generated by the initial forms of the elements of ${\displaystyle N}$.

Examples

Let U be the universal enveloping algebra of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that ${\displaystyle \operatorname {gr} U}$ is a polynomial ring; in fact, it is the coordinate ring ${\displaystyle k[{\mathfrak {g}}^{*}]}$.

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

${\displaystyle R=I_{0}\supset I_{1}\supset I_{2}\supset \dotsb }$

such that ${\displaystyle I_{j}I_{k}\subset I_{j+k}}$. The graded ring associated with this filtration is ${\displaystyle \operatorname {gr} _{F}R=\bigoplus _{n=0}^{\infty }I_{n}/I_{n+1}}$. Multiplication and the initial form map are defined as above.

See also

• Graded (mathematics)
• Rees algebra

References

1. Eisenbud 1995 , Corollary 5.5
2. Zariski & Samuel 1975 , Ch. VIII, a paragraph after Theorem 1.

• Eisenbud, David (1995). Commutative Algebra. Graduate Texts in Mathematics. Vol. 150. New York: Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN   0-387-94268-8. MR   1322960.
• Matsumura, Hideyuki (1989). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated from the Japanese by M. Reid (Second ed.). Cambridge: Cambridge University Press. ISBN   0-521-36764-6. MR   1011461.
• Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN   978-0-387-90171-8, MR   0389876