Let $(R,\fm)$ be a commutative Noetherian local ring of dimension $d$ with residue field $k=R/{\fm}$. Let $M$ be a finitely generated $R$-module. Let $I$ be an ideal of definition for $M$. Let $G_{I}(R)$ be the associated graded ring of $R$ with respect to $I$ and let $G_I(M)$ be the associated graded module of $M$ with respect to $I$ considered as a graded $G_I(R)$-module. Let $H^I_{M}(n)=\lambda(M/{I^nM})$ denote the Hilbert-Samuel function of $M$ with respect to $I$. The Hilbert-Samuel function is given a polynomial $p^{I}_M(n)$ of degree $\Dim M=r$. It is written in the form $p^{I}_M(n)=\sum_{i=0}^r(-1)^ie_i^I(M)(^{n+r-i}_{r-i})$. The integers $e_0^I(M), e_1^I(M),..., e_r^I(M)$ are called the Hilbert coefficients of $M$ with respect to $I$. The number $e_0^I(M)$ and $e_1^I(M)$ are also called the multiplicity and Chern number of $M$ with respect to $I$, respectively. If $I$ is generated by a system of parameters for $M$, then $M$ is Cohen-Macaulay $R$-mosule if and only if $e_0^I(M)=\lambda(M/{IM})$ and in this case $e_1^I(M)=0$. The existence of the the Hilbert-Samuel polynomial is equivalent to the fact that the formal power series $\sum_{n\geq 0}\lambda(M/{I^{n+1}M)z^n$ represents a rational function of a special type: $\sum_{n\geq 0}\lambda(M/{I^{n+1}M)z^n=\frac{h_M^I(z)}{(1-z)^{r+1}}$ and $h_M^I(z)=h_0^I(M)+h_1^I(M)z+...+h_s^I(M)z^s\in\mathds{Z}[z]$. The polynomial $h_M^I(z)$ is called the $h$-polynomial of $M$. If $f$ is a polynomial we use $f^{(i)}$ to denote the $i$th formal derivative of $f$. It is easy to see that $e_i^I(M)=h_M^{(i)}(1)/{i!}$ for all $i\geq 0$. I will study the Hilbert-Samuel function and the Hilbert-Samuel polynomial and also the multiplicity and Chern number of $M$ and $G_I(M)$ with respect to $I$.
