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 χ_M^I :N->N such that, for all n element N, χ_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 χ_M^I(n) = sum_(i = 0)^n H(g r_I(M), i). For sufficiently large n, it coincides with a polynomial function of degree equal to dim(gr_I(M)) - 1.
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