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