TOPICS
Search

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 Module

This entry contributed by Margherita Barile

Explore with Wolfram|Alpha

References

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

Referenced on Wolfram|Alpha

Associated Graded Module

Cite this as:

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

Subject classifications