Associated Graded Module
Given a module M over 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 module of M with respect to F is
| gr_F(M)=I_0M/I_1M direct sum I_1M/I_2M direct sum I_2M/I_3M direct sum ..., |
(2)
|
which is a graded module over the associated graded ring gr_F(R) with respect to the addition and the multiplication by scalars defined componentwise.
If I is a proper ideal of R, then the notation gr_I(M) indicates the associated graded module of M with respect to the I-adic filtration of R,
| gr_I(M)=M/IM direct sum IM/I^2M direct sum I^2M/I^3M direct sum .... |
(3)
|
See also
Associated Graded Ring, Hilbert-Samuel Function, Rees ModuleThis 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 ModuleCite this as:
Barile, Margherita. "Associated Graded Module." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AssociatedGradedModule.html