Hilbert-Samuel polynomial

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Definition

Let A be a Noetherian local ring with unique maximal ideal \mathfrak{m}. Let M be a finitely generated A-module, and I be an ideal of finite colength in A (in other words, there exists a n such that \mathcal{m}^n \subset I \subset \mathcal{m}).

Consider an essentially I-adic filtration of M, i.e. a descending chain:

M = M_0 \supset M_1 \supset M_2 \supset \ldots

such that there exists n_0 such that M_{n+1} = IM_n for n \ge n_0.

The Hilbert-Samuel function for this filtration is a function sending a positive integer d to the length of the quotient M_{d-1}/M_d as an A-module. This Hilbert-Samuel function turns out to be equal to a polynomial for sufficiently large values of d. That polynomial is termed the Hilbert-Samuel polynomial for the filtration.

Some people use the term Hilbert-Samuel polynomial for the length polynomial, which is the polynomial measuring the length of the module M_0/M_d.

When we simply talk of the Hilbert-Samuel polynomial, we by default refer to that for the ring as a module over itself. The default ideal is taken to be the maximal ideal and the default filtration is taken to be the standard one.

Facts

  • The specific Hilbert-Samuel polynomial depends on the specific filtration we choose. However, the leading coefficient of the Hilbert-Samuel polynomial depends only on the ideal I.
  • The leading coefficient depends on the specific choice of the Hilbert-Samuel polynomial. However, the degree is independent of the choice of I.
  • For a Noetherian local ring R, the degree of the Hilbert-Samuel polynomial equals the Krull dimension.