Projection formula

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

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 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 $I^{m}M$ 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.

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 $\mathrm{gr}U$ 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

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

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

See also

• Graded (mathematics)
• Rees algebra

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• 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.