Projection formula

In mathematics, the associated graded ring of a commutative ring R with respect to a proper ideal I is the graded ring $\operatorname {gr} _{I}R=\oplus _{n=0}^{\infty }I^{n}/I^{n+1}$. Similarly, if M is a R-module, then the associated graded module is the graded module $\operatorname {gr} _{I}M=\oplus _{0}^{\infty }I^{n}M/I^{n+1}M$.

Basic definitions and properties

For a ring R and ideal I, multiplication in $\operatorname {gr} _{I}R$ 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 $a'\in I^{i}$ 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 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 $I^{m}M$ where m is the maximum integer such that $f\in I^{m}M$. If $f\in I^{m}M$ for every m, then set $\mathrm {in} (f)=0$. 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 $\operatorname {gr} _{I}M$ 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 $\operatorname {gr} _{I}R$ is an integral domain, then R is itself an integral domain.

Examples

Let U be the 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 $k[{\mathfrak {g}}^{*}]$.

Generalization to multiplicative filtrations

The associated graded can also be defined more generally for multiplicative descending filtrations of R, Let F be a descending chain of ideals of the form

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

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

See also

• Graded (mathematics)
• Rees algebra

References

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• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
• H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.