The Hilbert-Samuel function and eventual Hilbert-Samuel polynomial of a finite local module

Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $M$ be a finite $R$-module, and let $I\subseteq \mathfrak{m}$ be an ideal of definition for $M$, meaning that $M\mathrm{/}IM$ has finite length. Every quotient below has finite length: its finite $I$-adic filtration has factors that are finite quotients of finite direct sums of $M\mathrm{/}IM$. The Hilbert-Samuel function of $(M,I)$ is

$${\chi }_{I,M}(n):={\mathrm{\ell }}_R(M\mathrm{/}{I}^{n+1}M)(n\ge 0)\mathrm{.}$$

The associated graded module is

$${\mathrm{gr}}_I(M)=\underset{n\ge 0}{⨁}{I}^{n}M\mathrm{/}{I}^{n+1}M,$$

and its homogeneous-piece lengths are

$${\phi }_{I,M}(n):={\mathrm{\ell }}_R(I^{n}M\mathrm{/}{I}^{n+1}M)\mathrm{.}$$

They satisfy

$${\chi }_{I,M}(n)=\sum _{j=0}^n{\phi }_{I,M}(j)\mathrm{.}$$

This is repeated additivity of length in the finite filtration $M\supseteq IM\supseteq \cdots \supseteq I^{n+1}M$.

When there is a polynomial $P_{I,M}(X)\in \mathbb{Q}[X]$ with

$${\chi }_{I,M}(n)=P_{I,M}(n)\text{for all sufficiently large }n,$$

it is called the Hilbert-Samuel polynomial of $M$ with respect to $I$.