Associated Graded Ring
Given a commutative unit ring R and a filtration
| F:... subset= I_2 subset= I_1 subset= I_0=R |
(1)
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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)
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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)
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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)
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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)
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See also
Associated Graded Module, Hilbert-Samuel Function, Rees RingThis 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 RingCite this as:
Barile, Margherita. "Associated Graded Ring." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AssociatedGradedRing.html