Hilbert-Samuel polynomial

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.