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Associated Graded Ring

Given a commutative unit ring R and a filtration

F:... subset= I_2 subset= I_1 subset= I_0=R
(1)

of ideals of R, the associated graded ring of R with respect to F is the graded ring

gr_F(R)=I_0/I_1 direct sum I_1/I_2 direct sum I_2/I_3 direct sum ....
(2)

The addition is defined componentwise, and the product is defined as follows. If a=[alpha]_i in I_i/I_(i+1) is the residue class of alpha in I_i mod I_(i+1), and b=[beta]_i in I_j/I_(j+1) is the residue class of beta in I_j mod I_(j+1), then a·b=[alphabeta]_(i+j) is the residue class of alphabeta in I_(i+j) mod I_(i+j+1).

gr_F(R) is a quotient ring of the Rees ring of R with respect to F,

gr_F(R)=R(F)/t^(-1)R(F).
(3)

If I is a proper ideal of R, then the notation gr_I(R) indicates the associated graded ring of R with respect to the I-adic filtration of R,

gr_I(R)=R/I direct sum I/I^2 direct sum I^2/I^3 direct sum ....
(4)

If R is Noetherian, then gr_F(R) is as well. Moreover gr_F(R) is finitely generated over R/I. Finally, if R is a local ring with maximal ideal M, then

dim(gr_(M(R)))=dim(R)=dim(R(F))-1.
(5)

See also

Associated Graded Module, Hilbert-Samuel Function, Rees Ring

This entry contributed by Margherita Barile

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1993.

Referenced on Wolfram|Alpha

Associated Graded Ring

Cite this as:

Barile, Margherita. "Associated Graded Ring." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AssociatedGradedRing.html

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