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Hilbert-Samuel Function


Given a nonzero finitely generated module M over a commutative Noetherian local ring R with maximal ideal M and a proper ideal I of R, the Hilbert-Samuel function of M with respect to I is the map chi_M^I:N->N such that, for all n in N,

 chi_M^I(n)=l(M/I^(n+1)M),

where l denotes the length. It is related to the Hilbert function of the associated graded module gr_I(M) by the identity

 chi_M^I(n)=sum_(i=0)^nH(gr_I(M),i).

For sufficiently large n, it coincides with a polynomial function of degree equal to dim(gr_I(M))-1.


See also

Hilbert Function

This entry contributed by Margherita Barile

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References

Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.

Referenced on Wolfram|Alpha

Hilbert-Samuel Function

Cite this as:

Barile, Margherita. "Hilbert-Samuel Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Hilbert-SamuelFunction.html

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